English does not have a Dative case (I have a strong feeling those Russian reflexives are usually Dative, not Accusative; please correct me if I'm wrong, but that's what I'd expect in German, for instance), and English reflexive pronouns are cumbersome Rube Goldberg constructions (viz, myself, ourselves, yourself, yourselves, himself, herself, itself, themselves, including the only 2nd person plural inflection in the language) which demonstrate that English does not use reflexives much at all for grammatical purposes.
Consequently, most of the sentences above are either ungrammatical (1 & 2; though they're understandable, and such unnecessary reflexives are easily ignored); an example of a wrong lexical choice (6; you can't feel yourself doing anything when you're sleeping - the word for that is dream); or part of a different construction that probably doesn't carry the same meanings as the Russian construction.
3, 4, & 5, however, are grammatical examples of the construction called B-Raising. Not to go into details, these reflexives are required after the verb feel (which means 'believe, perceive' in this construction) when it takes an infinitive complement clause with to that represents a belief or perception about oneself.
be a hero] ==> I feel myself to be a hero.
be an astronaut] ==> I feel myself to be an astronaut.
be suppressed] ==> I feel myself to be suppressed.
The reflexive is required only because the subject of an infinitive can't be deleted by Equi after like, so it gets Raised up to become the direct object of like, and then it has to become reflexive because it's identical to the subject of like. It's a very complex rule.