So, first of all, there are actually two different ways that the words "I feel myself" can be used in a sentence, and thus really two different meanings they can have:
- "I feel myself (verb)ing" essentially means "I feel the physical sensation of myself doing (verb)". In this case, "feel" is being used with a meaning of actual tactile sensation (i.e. the sensation of touch, or motion, etc).
- "I feel myself to be (thing)" or "I feel myself to have (thing)", however, means "I believe that I am/have (thing)". In this case, "feel" is just being used as another word for "believe" or "think".
(Also, note that sometimes the "to be" part is left out and just implied, so it can sometimes be said as just "I feel myself (thing)", but it means the same thing as "I feel myself to be (thing)")
So in your examples:
- I feel myself flying --> I feel the physical sensation of myself flying
- I feel myself to have a duty --> I believe that I have a duty
- I feel myself the equal of Moses --> I believe that I am the equal of Moses
The reason your first examples are not really correct is because they are using the "I feel myself (verb)ing" form, which means they're talking about tactile sensation, but the verbs your using are not things that actually have a tactile sensation associated with them:
- I feel myself thinking --> I feel the physical sensation of myself thinking
What does "thinking" feel like? Can you describe the sensation? For most people, they don't actually feel a different sensation when they're thinking than when they're not, so this doesn't actually make a lot of sense.
- I feel myself dreaming --> I feel the physical sensation of myself dreaming
Again, does "dreaming" have a particular feeling that you get when you're doing it? Many dreams may involve feeling things, but what you feel probably changes from dream to dream, there's generally no specific sensation one experiences for "dreaming" itself, which someone could identify as "what dreaming feels like".
Basically, using this construction automatically brings up the implied question: What does doing that thing actually feel like? If you can't actually answer that question, then it's probably not a verb you can use that way.