Is it possible to like someone who dislikes everything you say/do or dislike someone who likes everything you say/do? How?

Stop using superlatives and I can answer.

Try most instead of everything.

Of course no one can get along with such a creature.

Anything's possible, however in this case it's highly improbable.  People prefer to be around others who share similar interests with themselves.  This is generally how relationships form.  If a person is constantly disliking everything you say and do, you won't want to be around them.  Eventually you'll tell them "where they can go" (and possibly how to get there).
While we like to be agreed with, we also want people around us who aren't afraid to, occasionally, tell us if we screw up (slight dissimilarities).  A person who likes everything you say is called a suckup, and most peopel find suckups annoying.  Eventually they'll anger you so much that you end up wanting to knock them through a wall.

Dislike, yes. Fawning yes-men I find annoying. It's endearing in a dog but intolerable in a human being.

I don't think I could like someone who dislikes what I say and do either. Respect, yes.