Bone, laŭ ESPDIC (http://www.denisowski.org/Esperanto/ESPDIC/espsearch.php):
- senti sin : to feel
- senti : to experience, feel, sense, perceive
Mi imagus, ke se mi sentas ion, mi sentas ion fizike. Sed, se mi sentas min ion, mi sentas min ion emocie.
Se mi malveras, bonvolu diri al mi.
I've always taken it to be an emphasis on who is feeling what. I've never tried to get away with something like «Li sentis ŝin feliĉa» but I suspect that would be read as "he felt that she was/is happy." Leaving off the second pronoun seems to place the sensation hanging in the air: «Li sentas feliĉa» removes the sense of the happiness happening to anyone and it's just sort of being there. he felt happiness (somewhere in the area)
It's like the verb needs an object.
Just my thoughts, and they seem to help me with this.
Yes, it's reflexive. In English you just say it in another way, in Esperanto (which I know very well) it's used like in German (which might not help you). But they'll have to tell the learners about it! Well, it's still in beta, they will need some weeks.
I wrote a remark that the dictionary did not say anything about it.
Yes, because that isn't how English grammar works. As a native speaker of the language it took me quite a bit of effort to understand the reflexive, which is common in many languages, but not too often in English.
And since that is what you're asking: I feel happy is adequate. If the reflexive is called for then it would be,I feel happy, myself.
I didn't invent English, I will not argue its merits or lack thereof.