Museums have funds to acquire items for their collections, but (as most museums are public or non-profit entities rather than private companies) it is a fairly drawn-out process with a lot of hoops to go through. There would be a written collecting policy in place, a committee or Board approval process, etc. This varies by institution so you might have a curator with discretion to buy small items at auctions, but it is rare. It's more common for art museums to acquire pieces (more common than history museums, I mean), but even then - it's rarely at "market" rate and often accompanied by a public appeal for something major.

So, for the question you're asking - almost certainly not. A museum would much prefer to have the item donated outright or maybe on a long-term loan (which they would not be paying you for). And it would have to be of pretty significant historic value, or filling a gap in their collection, for them to want it at all - most museums have a mere fraction of their collection on display, and once an item is accepted into the collection the museum is basically pledging to properly conserve and store that item for ever, unless it's deaccessioned from the collection (which is, again, a long and drawn out process). But, if you have something you think a museum might want - you should ask! It may be just the item they want, and your donation could be tax deductible!

Source for all of the above is my personal experience - having worked in both art and history museums.

No, they wouldn't. At least I can't think of a single case of that happening. If you had something historic that you might think a museum would be interested, you would let them know. They would consider whether or not it matches their mission, and whether they could properly care for it. Then they would either offer to take it as a gift (potentially tax-deductible) or a loan (which depending on the object/s many museums are moving away from for several reasons).

The majority of museums aren't exactly overflowing with excess wealth, and most only display a small fraction of their inventories at any one time anyway, so there isn't much incentive for them to pay to display.

Now there is one type of instance where they would, and that is for a traveling exhibit. For instance, institutions paid to have the King Tut exhibit, which drew massive crowds. But I get the feeling that that wasn't what you were asking.

EDIT: See /u/cpatterson's comment about museum acquisition funds, which entirely slipped my mind.

OP, you might also try /r/MuseumPros