Needed a second opinion on these translations (mainly in regards to word usage/conjugation)

Hello, recently I was trying to translate some of the track titles from the Devil May Cry 3 soundtrack, but I was on the fence about a few of them. The bellow are all titles for cutscene BGMs and they're all simply named after whatever event transpired during said cutscene.

"塔の頂で見ていたバージルとアーカム"
Online I've seen this one translated as "Vergil and Arkham Seen at the Top of the Tower," but I think "頂で見てい" would actually be "Looking/Watching from the Top of the Tower" in this context. All of the other cutscene BGM titles appear to be written in a way that just describes the action or event taking place so it would be weird to use "seen" since it makes it sound like the scene itself is being viewed by an outside party. From what I can find "見てい" can be either "looking" or "watching" though I think it might make more sense to go with "looking" since "watching" implies that the two are looking at something specific which seems unlikely considering how high up they are, Arkham even comments "isn't it a magnificent view?" which seems pretty general. Thus, I'm thinking that "Vergil and Arkham Looking from the Top of the Tower" might be a more proper translation than "Vergil and Arkham Seen at the Top of the Tower."

"レディをキャッチ"
So I think this one would literally be "Catch Lady" (in this context "Lady" is a character's name as opposed to a general term for an unnamed women.) However, that sounds kind of weird in English, can "キャッチ" also be read as "Catching" in English depending on the context? I think that "Catching Lady" would sound a little better than "Catch Lady/Lady Catch" when said in English. The scene in question is pretty straight forward, the character Lady was thrown off of the tower and the protagonist Dante who was on a lower level of said tower catches her (by grabbing her leg) as she's falling.

"永久機関セット (ダンテリフティング)"
I know that "リフティング" literally means "lifting" in English, though it can apparently also mean "juggling" in certain contexts which is what is occurring in this scene. Basically I was curious to know if "リフティング" would exclusively be "lifting/juggling" in English or if it could also be read as "lifts/juggles." Also as a side note, would "Eternal Mechanism Set" be a proper translation for "永久機関セット?"

"消えたアーカム"
I think this one would literally be "Arkham Disappeared/Vanished," I think "消えた" is exclusively past tense, but I was curious to know if this could be read in present tense or not since I thought "Arkham Disappears" might sound better if either can be used, if not though I'll probably just go with "Disappeared" or "Vanished."

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There were a few other ones that I was uncertain about as well though not specifically in regards to conjugation.

"レディの排除に向かうアーカム"
Online I've seen this one translated as "Arkham Faces Lady's Rejection" which just seems like a word salad to me. In the scene in question Vergil states that someone else (Lady) is in the tower and Arkham says that they'll "ask them to leave" presumably meaning that he intends to kill them. Also, as mentioned before "Lady" in this context is a character's name. I was thinking maybe something like, "Arkham Goes to Eliminate Lady," would that be an acceptable translation for "レディの排除に向かうアーカム?"

"橋落下"
I'm aware that this one literally means "Bridge Falls," but I was curious if "落下" could also be read as "collapses" or not since whenever talking about a bridge coming down I almost always see it described as a "collapse" the only noteworthy example I can think of where "fall" is used instead is in the nursery rhyme "London Bridge is Falling Down." Either way, if it is exclusively "fall(s)" then I suppose that's what I'll probably end up going with.

"父娘決着"
This is more of an opinion based question, basically I couldn't decide what sounded better for a translation "Father and Daughter Conclusion" or "Father-Daughter Conclusion," I think the latter may sound a little less stilted. For context, the title is referring to this being the scene where the Arkham and Lady (who are father a daughter) subplot is wrapped up.

I'll take a stab, although we have a couple of posters here who are ten times better than me at describing grammatical structures. (Where are you bentenmusume?)

1. In this case, "塔の頂で見ていた" is an adverbial phrase modifying Vergil and Arkham (Vergil and Arkham at the top of the tower). The sentence doesn't say who was doing the seeing.
2. I prefer "Catching Lady", as it feels more colloquial in English. "Catch Lady" in English is more like an imperative, which adds a nuance that isn't in the original. So both of your English options are imperfect matches for the original Japanese, but to my mind the former is so close to the original as to be identical, while the later takes a step too far away from the original.
3. Eternal Mechanism Set (or Eternal Machine Set... I don't know what the original story implies). "Lifting", as in soccer-ball lifting, is the same in English. It is a kind of juggling of the soccer ball using only your feet, so lifting is a specific kind of juggling, and the words are not completely interchangeable. In this case, as in the previous case, the present participle ("lifting") feels colloquial and natural to me, and is aligned with the original, but the third-person present is also valid.
4. Depends what the scene is. Is it a scene where he exists and then disappears, and the disappearance is the substance of the scene? If so, I would say that "Arkham Disappears" is valid, even though its a significant grammatical shift from the original. The original emphases Arkham (the Arkham that Disappeared), while in English "Arkham Disappears" emphasizes what is happening to Arkham. So it sort of changes which word is doing the heavy lifting, however the difference in this case may be fairly immaterial, and so Arkham Disappears might be the right choice, all things considered.
5. Yes, your translation is fine. Other options might be "remove" or "expel".
6. When talking about a bridge, or any structure, its find to translate 落下 as "collapse" or "collapses".
7. Yes that's fine.

Hey there, thanks a lot for the reply, it was very helpful. If I may, I had some follow up questions/comments regarding some of your responses.

"塔の頂で見ていたバージルとアーカム"
Just to clarify, you're saying that the way this one is written out would actually be read as "Vergil and Arkham at the top of the tower" as opposed to "Vergil and Arkham looking/watching from/seen at the top of the tower" correct?

"永久機関セット (ダンテリフティング)"
In regards to the first half I'll attempt to explain the context as best as I can. There's a key-item called the "Neo-Generator," the description of said item is "A magic artifact able to supply power for an eternity. It is used to power the main bridge." At the rotating bridge Dante beings to juggle the Neo-Generator (which is spherical) like a hacky sack or soccer ball and then kicks into the slot above the bridge in order to power and turn it. Here's the scene for context if that helps any. Considering the mention of "eternity" in the item description and the specification that something is being "set" (セット) I think it's the Neo-Generator that's being referred to here as opposed to the machinery that operates the bridge. I don't really think it qualifies as "machinery" so "mechanism" is the only term I can think of that would make sense since that can be used to refer to a part of a machine, though I'm uncertain if what essentially amounts to glorified battery constitutes as a "mechanism."

As for the second half, I'm aware that "リフティング" ("lifting") is a type of exercise where you juggle a ball with your foot and or knee. However, I've personally only seen this term used for that purpose in Japanese. In English whenever someone says "lifting" the first thing I imagine they'd think of is weight lifting. Most English examples that I could find, such as this one seem to just refer to it as "juggling," I even found a Japanese video which labeled it as "juggling" in the title. Admittedly, when one hears the term "juggling" they'll likely think of "toss juggling" specifically, but juggling can also refer to things kept in the air with your feet as opposed to your hands. I'm reminded of the game, Rhytmn Heaven which had a level titled "リフティング" in the original Japanese version, but in the English release the level was instead localized to "Space Soccer" probably for the same reasons I just mentioned. Though regardless of which I use I'm still undecided on which sounds better "Dante Lifts/Juggles" or "Dante Lifting/Juggling." Every exsample I've seen for "リフティング" thus far has used "ing" instead of "s" so I'll probably end up going with that spelling.

"消えたアーカム"
At the start of Mission 11 the character Arkham is stabbed and seemingly dies, however during the brief cutscene at the end of that same mission (this being the cutscene which this title corresponds too) his body is no longer there. Since you don't get see his body disappearing onscreen it must have happened at some point between the start and end of that mission so I guess that means it would technically make sense to use past-tense in this scenario, though something about "Arkham Disappeared" still sounds a little weird to me as a title.

"橋落下"
Just out of curiosity would you by chance know of any examples where "落下" is expressly used for "collapse" instead of "fall", certainly in this specific context it makes more sense, but I was having difficulty finding examples where it was clearly the intended meaning.

"父娘決着"
Just to clarify, were you saying that either translation would be fine? If so which of the two do you personally think sounds better?

1. Vergil and Arkham (as seen) at the top of the tower.

"Vergil and Arkham seeing (something) from the top of the tower" is a slight possibility, but it feels like a very remote possibility since the direct object is missing, and makes the sentence very tenuous. "Watching" would be a different verb: 見つめる or 見張っている

(help me, Toritoribe-san!)

2. I still can't tell what 永久機関 is, but yes its reasonable to assume its describing the "setting" of the bridge mechanism. For the second part, it feels like we're both describing the verb "lifting" to each other. I actually thought this kind of "lifting" was more widely understood outside of Japan, but if it isn't well understood outside of Japan, you may want to use a different verb. To me, juggling doesn't work because of that verb's strong association with toss juggling, but your mileage may vary. In a situation like this, depending on your confidence level and your client's trust in you, you might want to completely abandon the literal translation and go for something like "Dante Scores A Goal" which conveys the playful soccer-like kicking, without confusing the reader with a word like "juggling" which somehow seems to lead the western reader in a weird direction.

3. Don't worry too much about faithfully reproducing the past tense or present tense. In Japanese the perspective and use of tenses is not entirely synchronous with English (there is an amazingly long thread on this here on the JREF forum). "Arkham Disappeared" is fine and valid, as is the present tense, "Arkam Disappears".

4. 橋が落下して２人死亡、負傷者も フランス

5. The hyphenated version sounds fine, and better, in my opinion. This is a matter of style. Over-reliance on punctuation can make things difficult to read, but in this case I think the lack of the hyphen somehow makes the Father-Daughter pairing feel a bit lonely.

As bentenmusume-san already pointed out in his post previously, it's impossible to know the meaning of words correctly without knowing the context in detail. For instance, リフティング can refer to an action "lifting/rasing something up", and ダンテ can be both the agent and object in that phrase, i.e., ダンテリフティング can mean both "Dante lifting/raising something up" and "something/someone lifting/raising Dante up". I can't judge even whether this interpretation is valid or not since I don't know the context.

Also, the translation can differ depending upon how literally you want to translate the original Japanese. For instance, 塔の頂で見ていたバージルとアーカム is a noun phrase. "Vergil and Arkham looking/watching from/seen at the top of the tower" might be natural in English with keeping the structure as a noun phrase, but it loses the tense which the original Japanese has. In fact, this English translation can't express the difference between 塔の頂で見ていたバージルとアーカム and 塔の頂で見ているバージルとアーカム. Whether or not it's somehow unnatural, you need to use "Vergil and Arkham who were looking/watching from/seen at the top of the tower" in the perspective of the correctness about the tense.

Anyway, I can't be helpful since I don't know the story.