'I feel/I'm feeling' is automatically understood to be reflexive, and you don't need the 'myself,' себе/себя particle. You can just say: 'I feel good.' 'I'm feeling good.' There's not much difference in meaning here between the two. The first is more simple and declarative, whereas the second is more emphasizing your continual good mood/good state of health, etc.

The first two don't make sense. When you add a reflexive pronoun to the verb "feel" it completely changes the meaning. If you say that you "feel yourself", it means that you're using your hands to physical touch some other part of your body. When you say "I feel good", "good" is an adjective. If you say that you "feel yourself", that construction doesn't take an adjective.

Without "myself", your basic sentences are 1. I'm feeling good. 2. I feel good. The purpose of the word "myself" is to add emphasis to the word "I". You can put it almost anywhere in the sentence with no change of meaning: Myself, I'm feeling good. I myself am feeling good. I'm feeling good myself. Myself, I feel good. I myself feel good. I feel good myself. The one thing you cannot do is put "myself" after "feel". If you say "I feel myself" it means that you are literally feeling yourself. For example, you could be holding onto your elbow that you just fell on and feeling it to see if there are any broken bones. Also, "I feel myself" has some meanings that are not suitable for discussion here. In general, it is better to place "myself" right next to the word it modifies ("I"): "I myself" or "Myself, I..."

"I feel good" is the correct one. Alternatives include "Me - I feel good" and "I feel good myself". But the other two options you gave have a vulgar meaning. They have bad grammar if the vulgar meaning is not intended.

I have to be careful what I say so Italki won’t censure me, but ‘to feel oneself’ in English also has another ruder meaning, so I always tell my students not to say it. If you say ‘I feel’, there’s actually no need to then add the word ‘yourself’ too. It’s superfluous. Starting with ‘I’ clearly indicates you’re talking about you/yourself. I feel good.