Grammar questions in this sentence

長年の沈黙を破りついに復活。

I have a few questions (again : /).

First, I'm under the impression that the verbs (in masu stem form) can connect clauses just like the て form can so is it correct to assume that this sentence is constructed as 長年の沈黙を破り + ついに復活 (Year long silence broken + finally revived)?

If that's the case then what is the adverb (ついに) modifying? Is it attached to the preceding verb or is it somehow modifying 復活? Which is kind of odd to me since that's a noun, maybe there's some adverb rule or lack thereof that I'm not aware of?
I saw a lot of cases on google where where ついに is next to nouns without any verb at all like "ついに発表!" or "ついに登場", also a sentence from alc "静まりついに（人）は平安を得る" which I can't really figure out.

Or am I just totally wrong about all of this?

Yes the te-form or masu form is called 連用形 in Japanese and is the 2nd form of verb conjunction. As the word implies, it is used to join.

Both 破り and 破い is valid.

ついに is an adverb. Adverbs precedes the verb it modifies unlike English which is the opposite. In this case it modifies 復活. Resurrected at long last.

Just like 速く走る run quickly. 丁寧に梱包する packs neatly/carefully/with care.

In your last sentence, ついに is modifying 得る. To obtain at long last.

Also ついに登場、発表 etc, these aren't nouns, well they are but in this case they are サ変 verbs with the 送り仮名 chopped off. This is common in headlines as it gives more impact as evident by Japanese newspapers. The full sentence should be ついに登場する！but the kana just get in the way so the drop it.

Newspaper do this all the time. Eg
新藤総務相が靖国神社参拝 中韓の反発強まる可能性

If you put all the kana in it would be
新藤総務相が靖国神社を参拝することに 中韓の反発が強まる可能性がある

Less impact, doesn't add anything, waste more ink and space so they just drop obvious particles.

Awesome, this helps so much. Thank for the info. :>