No, it is not correct English to say "I feel painful."
You can say "I feel pain", or "I am in pain", because 'pain' is a noun, but "painful" is an adjective so you would need to use it qualify something else (as in your "I feel painful sensations" example).
All of these sentences are grammatically valid, but for some of them the intended meaning is not at all clear, and they are not the way that most English speakers would express these ideas.
In general, "I feel myself" is generally understood to mean touching yourself for autoerotic pleasure, which is probably not what you mean in any of these examples.
"I feel myself badly." Sounds like you mean that you are unskilled at autoeroticism. If what you mean is that you are sick or unhappy, you should say simply "I feel bad."
"I feel myself well." Similar to badly but in the opposite direction. You probably mean "I feel good."
"I feel myself to be a hero/astronaut/suppressed." Valid. These would be understood to mean that you think you "qualify" as one of these things, but by using the word "feel" rather than simply stating that you "are", you imply that the classification might be debatable. Like, someone who has flown very high-altitude airplanes might say, "I feel myself to be an astrounaut", knowing that others will challenge the claim. I think most Enlgish speakers would be more likely to say, "I consider myself to be an astrounaut" or "I think of myself as an astronaut" or "I think I am an astronaut." But the sentence as written is valid if that's what you mean. Note this is different from saying, "I feel like a [whatever]". In that case, you are not claiming to actually be whatever, just that you have some similar experience. Like, "After an hour in the Space Shuttle simulator, I feel like an astronaut."
"I feel myself sleeping." I'm not sure what you're trying to say. If you're sleeping, you're not really feeling anything. Maybe "I feel myself falling asleep"?
Best Answer
Assuming you're talking about a situation where something bad has happened to your friend, and you're saying you feel unhappy on their behalf, then "I feel bad for you" is correct.
You are right that "I feel badly" would mean you are having difficulty in feeling at all - which would be a rather unusual thing to say :)
In general the verb "to feel" will take an adjective (happy, sad, good, bad, angry, relieved) after it to indicate the feeling, rather than an adverb.