It sounds like you are reading literary text, so the word usage is going to be a bit uncommon.
To "prove one's innocence" means the person is proving their own innocence. "One's"~"One's own". You might say: "In order to prove one's innocence, one should provide all the facts truthfully". Again, very formal.
attach, fix, and fasten I think are all slightly different, but are very similar. Attach is the most common and general. I think fix and attach mean about the same thing, but I'm not certain. This use of "fix" is quite rare though. Fasten is also relatively uncommon, and refers to things like ropes, chains, and seat belts.
i think "onto" can usually be replaced by "to", like we could say "fixed onto...". But onto is more specific. It means the result is "on" whatever, while "to" is more ambiguous. In some cases "into" can also be replaced by "to", but not as often. "into" means the result is "in" whatever, so I don't think you can ever replace "onto" with "into". Examples: He ran into the house. (Now he's inside) He ran to the house. (Now he's next to the house). He ran onto the house. (Now he's on the roof. A little strange. Almost sounds like he was running on top of houses or something.)
rim, edge, and side are a bit different in usage. Rim is a very specific word for like wheels. Edge usually refers to a horizontal boundary (but you're still on top). Like edge of a nation, edge of a cliff, edge of a table. It can also refer to sharp things like a knifes edge. Side has a lot more meanings, but for example the side of a cliff is what you would climb.
Finally, this last one is almost poetic, but the answer is yes you can replace it. It will make it more specific, and less poetic, and doesn't fit as well with "the city" used in the second sentence. The author is focusing on the essence of the country and the city, which is deeper than just living in that place.