Learn English – Does erudite carry positive or negative or neutral connotation

"See if you can't" and similar expressions are a slightly old-fashioned but still common idiom in British English.

I cannot find any definite difference in meaning from "see if you can", except perhaps a hint of the vagueness that is used to add a little politeness (like "would you be able to?" for "will you?").

Edit: missed your last question: no, this is limited I think to verbs of finding out ("see" in that sense, "check", "find out", "work out"), uncertainty ("wonder" and perhaps "not be sure") and permission ("ask" in the sense of asking permission). I don't think it's used with "doubt", "tell" or "know", or with "ask" when asking about information rather than permission. It also sounds odd with more formal verbs like "verify", "determine" or "ascertain".

"Doubt" is an oddity, because it would seem to fit the category of uncertainty, but "I doubt that/whether he won't go" doesn't sound right to me.

The phrase means that everything went well until now, and you are not sure if things will carry on like this, or something bad will happen in the future, but it doesn't carry a negative connotation. It just means that so far everything is good, and you don't know what is going to happen next.