Just curious, does "Would you mind and do something" instead of "Would you mind doing something" sound acceptable in spoken AE, or is it an attempt to imitate or render colloquial speech in not so formal writing?
According to my ear, there's nothing wrong with those types of expressions. (I'm an AmE speaker, and I've traveled to or lived in most corners of the USA.)
Actually, in my personal opinion, I'd consider those versions to be more polite than these alternatives:
as those last three versions have the speaker sorta already assuming that the other person won't mind hearing a request from him, and the speaker is rushing a request at the other person. (Though others might disagree.)
As for the OP's #1, #2 and #3 examples, I've heard these kinds of phrases spoken often by AmE speakers, and I'm pretty sure I've used them myself. E.g. "Would you mind and do something for me?"
I'd consider these kinds of expressions to be at least of an informal style of standard English. The coordinator "and" has many uses, e.g. "Be sure and lock up", "We always try and do our best". (There might be some related info in the 2002 reference grammar CGEL.)
"Would you mind and do something" is unacceptable in any region of the US in which I have ever lived (Northeast, Mid-Atlantic, South, Mid-west and Northwest.) It is not idiomatic, and at best is a terrible example of attempt at simulating non-native speech.
Googling "would you mind and do" turns up no examples of this phrasing except your own question. The most common hit is for two polite phrases
We do have bizarre idiomatic speech pockets (as with PA Dutch), but none quite that bizarre.
I think that's either a transcription error or extremely non-standard.
mind and provide don't functional as separate verbs here. The main verb is mind and providing the phone number is the clause that is being asked about.