1-You'll get people coming up to you and asking for your autograph.
2-You'll have people coming up to you and asking for your autograph.
3-You'll have people come up to you and ask for your autograph.
Are all these sentences correct? Do they imply that it is bad to have people come up and ask for autographs? In other words, do these structures have a negative connotation, or could they simply mean "experience" in a neutral sense?
They're correct. They could be neutral or positive\negative.
I'd say the third is less likely to be negative.