As the others have pointed out, there isn't any one word that covers all those meanings.
- More readily; preferably: I'd rather go to the movies.
Prefere can be used in this context, but I find it flows better to use preferi as a verb: Mi preferus iri al la kinejo.
- More exactly; more accurately: He's my friend, or rather he was my friend.
I agree with the others, that "pli ĝuste" is the most suitable expression for this context.
- To a certain extent; somewhat: It's rather cold out. I was rather hoping you'd call.
"Iom" works here; another word used to mean "rather, quite" is "sufiĉe".
3 Mezgrade, tolereble, konsiderinde: li rimarkis en sufiĉe granda distanco du rajdantojnZ; sufiĉe gaja vivovesperoZ; cetere ĝi (la muziko) estis sufiĉe bonaZ; sufiĉe stranga estis tiu kostumoZ.
ReVo advises against this usage, but it is quite common. Note that it is generally used in that sense to modify adjectives and adverbs, not verbs directly. Thus "Mi sufiĉe esperis, ke vi vokos min" sounds strange. For that I would use "iom", or, if the intent is to emphasize that you were indeed hoping for a call, I'd say "ja".
- On the contrary: This is not a thoughtful criticism. Rather it is an insult.
One could use "pli ĝuste" here, but "kontraŭe" or "male" are more suitable.