In your first sentence, the "felt not" options are wrong. They would have been acceptable hundreds of years ago, but the simple past is now "did not + V"; 'he did not feel like being a victim' would be wrong in most contexts (although it would be OK in a self-defence class, for someone who, in practice, was always an attacker and wanted a change ). So, of your options, only one is right - 'he did not feel a victim'. There are alternatives though: 'he did not feel [as if] he was/had been a victim' - and using a verb after 'feel' avoids the "hamburger/idiot" ambiguity.
In your second sentence an impersonal "it" would help: "it felt [to him] as if everyone was his friend..." .
In your second sentence 'felt like being' would only be acceptable in an unusual context in which they felt like being [i.e. wanted to be] those things: 'He felt like being my friend' means 'he wanted to be my friend'; it doesn't mean 'I felt that he was my friend.' As in the cases of 'he felt like [being] a victim' and 'I feel like [eating] a hamburger' this use of 'feel like' denotes a preference for some future action or state.
In that sentence, 'felt being' is wrong; 'felt to be' is OK, but it isn't clear who's doing the feeling - so the version with 'it felt...' is preferable.