1.- What does the ~게 in 못하겠어 mean? And why 뭣 같아서 was traduced as "it sucks"?
겠 (게 without ㅆ is an entirely different element) has varying usages, but the underlying idea is to soften the meaning, similar to "would" or "might" in English. Simple 못해 would be a clear and conclusive "I can't", while 못하겠어 is like "I can't seem to" or "I'm trying but I can't".
뭣 (무엇) normally means "what" in a question and "something" in a statement, but it's also used as a slangy reference for something bad, like "you-know-what" in English when it's used as a euphemism for something which might be too offensive to put in words. So 뭣 같다 or 뭐 같다 can mean "It sucks".
2.- What does the 안대 in 네가 한 모든 말은 안대 mean?
안대 is short for "안다고 해" = they say they know - "-다고 (말/얘기)해" is a common way of quoting someone else's statement. However, this sentence as written is somewhat unclear because it doesn't say who is supposed to know everything you said (you don't use -ㄴ대 for yourself).
3.- What are the difference between 다, 모든 and 전부?
* 다(n, adv) - all or everyone/everything. It's an all-around word seeing especially heavy usage as an adverb.
* 모든(determiner) - all" or "every" (only used before a noun). A similar word 모두 is a noun as well as adverb.
* 전부(n, adv) - more formal sounding term similar to 모두 in meaning, but it's used more as a noun than adverb.
So 모든/모두 focuses on all members or things in a group; 다 emphasizes including all and not missing any; 전부 is like a more formal version of 모두, focusing on the group itself.
"모두 다" and "전부 다" are also common - they emphasize each and every one of group - e.g. 올 사람은 모두/전부 다 왔다 (Everyone who's supposed to come is here).
4.- What does 혼자 mean? And 혼자서만?
혼자 as a noun means an unaccompanied person, or "alone" as an adverb.
-서만 means "only/solely as" (서 = in a certain state; 만 = solely), so 혼자서만 means "only when I'm by myself", "keeping to oneself", "only as a loner", etc.