I work in the flooring industry & it is very common for baseboards to be installed prior to carpet going in (new builds & renos). Baseboards are commonly installed at 3/8" above the substrate to allow for pad & carpet thickness (unless you have a thick shag, or a thin commercial glue-down style carpet- adjust accordingly).

I personally would want the painting done first. I have been slowly painting the rooms in my house and no matter how careful I am, I have still gotten paint on the carpet! I will have to replace my carpet next year.

carpet first baseboard last ,the carpet needs to be under it . Paint the baseboards separately when dry have them installed ,

Subfloor, then baseboard and then carpet.