It's pretty late and after drinking on the way home I might not be in the right mindset to answer all of your questions, but let me try to hit on as much as I can. My apologies in advance if some explanations aren't perfect.
Zizka said:
(3)と I'd say to express "Subway AND bus". I wanted to put で for both (3) and (4) but I decided against it.
It's good that you decided against this. Simply doubling up on the same particle is not generally the way you express this. Just like if you wanted to say "I ate sushi and tempura." it would be おすしとてんぷらを食べました, not おすしをてんぷらを食べました.
Zizka said:
Could I get a hint for (5) please?
Do you have a tip when it comes to approaching compound particles? They seem to intimidate me a bit.
Compound particles as a general concept are more complex, but this particular example is fairly simple, as （ ）ついて (I won't explicitly state the particle because you say you only wanted a hint) is almost a set phrase/compound particle in itself. In fact, if you look it up in your Dictionary of Basic Japanese Grammar, you will probably find this as a single set construction.
Zizka said:
(3)(4)での, I had never seen that combination before. Not sure about this.
Yes, this is a slightly more complicated construction that I suppose it's understandable that you'd struggle with at first.
You understand that で can mark a location of action like 日本で生活しています ("I'm living in Japan."), right?
日本での生活 is a noun phrase meaning "my life in Japan", where 日本で is marked by の to modify 生活.
You could also find similar patterns like 銀行での仕事, 大学院での勉強, カナダ人としての経験, and so on and so forth. Compound particles are a tougher skill to master, so I'd encourage you to keep an eye out for further examples.
Zizka said:
(3)で: 浴びて means to wash oneself. I'd say the manner here, as in "wash yourself how?" by taking a shower.
This is something you just need to memorize/understand. With "to take a shower", シャワー is understood as the direct object of あびる. Incidentally, the corresponding phrase for "take a bath" is お風呂に入る (not お風呂を～）.
Zizka said:
(2) で where the action takes place. Here the verb is あります as in we eat... so it's not really an action. I think this one will be wrong but I can't think of anything else.
Possessive again の
Again, remember, it's not just the meaning of "location" that it's important to, but what it connects to.
うちで朝食を食べました will be で, because it's marking the location where the action 食べました is performed, but here there's no verb, it's just connecting directly to 朝食, which means it needs to take の. Incidentally, the compound particle うちでの朝食 would not necessarily be wrong, but で alone cannot be used to simply connect two nouns.
For (5) 〇〇の他に means "other than 〇〇". This is something where you just have to learn what 他 means and how it's used.
For (7) and (8), ～と～が～ isn't necessarily invalid. The implication here is that it's giving an example of what they eat, not explicitly stating this is what they eat and nothing else. Like the previous sample of the karaoke bar, I would argue if they want to explicitly state that this is in addition to something else, they might want to make that more clear. That said, considering that ゆで卵 and 野菜 clearly occupy a different group than the food already introduced, it makes sense that they'd be introduced with も (_also_ eggs and veggies, in addition to the rice and miso soup we already mentioned, rather than ～が, which suggests that's it and nothing else.)
Zizka said:
(2)を: I'm not sure. です is a copula. Do copulas ever require direct objects? It's the only particle which I can think of here. In fact, I'm pretty sure this is wrong.
This one was が, as new information I guess. :"New information" sounds vague to me. I mean, everything is new information in a sentence when you think about it.
It's best that you learn Xが好きです as a pattern. I know that in the English "I like X", like is considered a verb and X the direct object, but that doesn't mean the similar Japanese construction functions the same way. 好き is not a verb, but a な adjective, and in a standard sentence, the thing you like is marked by が, not を.
Zizka said:
Nope. So I was on the right track for だけ, the other one was a contrastive は. I thought だけ wasn't use for negative sentences.
Not sure where you heard that だけ is invalid for negative sentences. When used to suggest contrast, は (or any compound particle including は) is typically going to be valid. Again, much like 好き above, 食べられません (which is a potential verb, not a standard verb), will more likely take が (or は in the negative) than を.
Zizka said:
(8) adverbial. I looked at some sample sentences in the dictionary to figure this one out.
が new information I suppose. I still don't get that "new information" function to be honest.
This means "the school is nearby, so..." In that sentence (both the Japanese and English). "the school" is not serving an adverbial function.
Zizka said:
A particle modifies what precedes it. The noun clause preceding the particle is a time, so logically it's the time when the action takes place... but I missed the うち right at the very end unfortunately.
うち is not a time.
If it were 7時（ ）うちを出ます, then に would be a valid option to fill the blank, but that's not what we have here. (But perhaps you've already noticed that.)
h boy, another compound. There's an "only" here so perhaps だけ. Here however the sentence is negative and the meaning is of: only this and nothing else so I'd say しか. That's one of the particle. What about the other one however? I think this is wrong but を as the direct object of eating. 
