+ As of right now, i think what Doc need is improved Pills, and more hitstun on some moves ( Most notably Uair & Dtilt )
It's true that you can try to go toe-to-toe with Yoshi once he's close enough, Bair is so fast & powerful & has enough range to beat most of Yoshi's Aerials. For combos,you've got to analyze quickly if Yoshi's reaction is to Nair you or to double jump away, depending of the reaction, you can get different things going on.
About Eggs, just respect them. If you get hit by one of them, no worries, its unlikely that Yoshi will get much of it, or even leave you in a disadvantage state.