”See/wonder if +(a negative)” vs. “See/wonder if + (a positive)”

"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.

I agree with the comment above that "see if you can't" isn't outdated, at least not in the mid-Atlantic coast of the US.

Wonder/see if... is neither grammaticalized nor figurative (exactly) in my experience. In my experience each is being used fairly literally. The seeing in question isn't sensory, but it is very real. It is a finding out; likewise, the wondering is a expressive of real question.

The use of positive or negative clauses following the wonder/see if... is doing very much what it does in an interrogative. Asking a negative question implies the questioner expects a positive answer, and by asking a positive question the questioner implies that expects a negative answer.

My sister recently asked her three year old, "Didn't Mommy, Daddy, AND Uncle Ryan each tell you not to climb on the couch?" Clearly my sister knew that all three adults had, and expected her daughter to know and answer the same. A half hour later, "Did I tell you that you could have a cookie?" when my sister knew darn well that she had denied that selfsame request because dinner time was approaching. She expected her daughter to know and admit as much.

The same thing happens with wonder/see if... type clauses.

...wondering whether that hadn’t been Cedric’s plan...

The hadn't implies a positive answer, but the wonder mitigates its strength somewhat, perhaps. Harry suspects, but is not sure, that Cedric intended the noise to be heard.

These questions, especially the positive ones, can also be asked in such a context that they do not imply a presumed response. One might ask a passerby whether he has seen one's friend. There is no necessary supposition in such a case. On the other hand, the negative clauses almost always imply a supposed response. If you ask a passerby whether he hasn't seen your friend, you are certainly implying that you suppose he has, or at least should have, seen your friend.

The same does go for a number of other verbs. I imagine any verb intended to elicit a response or satisfy a desire for information can be used in this way. Ask certainly can. The real point isn't which verb is used to solicit information, but the solicitation of information itself. A declarative sentence can be asked verbally with a questioning tone and become in effect a question. In writing this is noted with a question mark, and positive clauses remain neutral or elicit a negative response; negative clauses elicit positive responses.