Coadjoint orbits need not be simply connected in general. I am not sure if this helps under your particular assumptions though.

oops! of course. So i guess my question was a little naive .... thank you anyway!

@Marco Farinati if i understand correctly, your comment essentially says that if it has a counit then it has (non-trivial) fin dim representations. So, algebras not admitting fin dim reps cannot be hopf. Have i understood correctly or am i missing something?

you are welcome. And .. welcome to MO!

yes, that is right.

I suggest you ping user @Paul Gilmartin on this question. He has various interesting works on connected HAs and maybe (?) he has something more to say on this.

In general no, since $A^\circ$ maybe trivial while $B^\circ$ may not.

I do not know a concrete example but a (somewhat more refined) version of your question is listed as an open problem (see: question 4.5.3, p. 59) in the following Phd thesis: dc.uwm.edu/cgi/viewcontent.cgi?article=2733&context=etd