One Day, You Stumble Upon A Box...

I would do the first or second options - it depends on what the coins are. I'd probably keep them all.

Keep some, sell some, and give some money to charity from the sales

I would keep some of it, use some of it to spark the interest in others to start their own collection. I would also donate some to local elementary schools to use doing history lessons so that the kids could see the history rather than just reading about it.

I voted other. I would just pass by and not get involved.
I'm too suspecious and would think something is wrong here.
As an example in real life some years back a TV station had a reporter stand on a corner and try to give away $1 bills. Almost inpossible to do. Can't remember the details but very, very few bills got handed out.
Amazing how so many people think they would grab money if it happened to them but when it really does, no one bites.

Funny thing happened to me just like "just carl" was saying. One day I was looking for some halves and my bank said that they had some that were "messing up their books" and they had to get rid of them and they'd give them to me for nothing. Well right away alarm bells were going off (it only amounted to $4) but every scenario I could see the end result for me did not look good. I ended up taking them but all the way home I kept looking in my rear view mirror looking for the police car with flashing lights. After all, how could I prove that the bank gave them to me, and who would believe me if I was caught with them. I was really sweating bullets and saying to myself that I was an idiot and everything else.

Anyway that was what was going through my head. Nothing came of it, but yes I was worried because no one especially gives away money for nothing. The cool thing was that there were 2 40%ers in the bunch. The reason that they were looking to get rid of the coins was that they were found money and they couldn't balance their books.

The whole thing was bizare to say the least.

But if the sign said free (and I'd take the sign to) they were meant for me and home they'd go and I would say a prayer of thanksgiving for my luck and to the goodness of the person who put them out. But before I'd go, before I left I would hold the sign up to the camera and wave adieu.

Put it in the car, then try to run away before the owner of that car sees me.

Alternately, using the only realistic was there could be a "my car"... I can easily imagine how could I manage to carry that heavy box! I would gently - so that the car isn't crushed - put it on the car, and then try and push the car all the way to my home - the wheels would help with this a lot. Then give the car back to my younger brother, put the box somewhere in the house and forget about it.
Several years (or possibly months) later, my brother accidentally finds the box, spends all the spendable coinage (probably not much since presumably most of the coins are American) and shows some of the others to me. I then say something along the lines of "1856 small cent? it couldn't possibly be real... probably someone's joke" and try to sell most of it on Taganka because I don't want to be stuck with a ridiculous amount of copy coins.
If I'm lucky, the expert at the Taganka shop is around, and he could say that the coins are (presumably) real. In that case, I would keep the obvious rarities because I don't know anybody who could afford them, and sell off the rest at some decent prices (probably give away a bunch of the less rare ones to good friends as well). Otherwise, I'm probably selling them off for five bucks a piece as copies.
Any way, the money I get from that goes partly to buy more coins and partly to whatever good-cause people I could find on short notice. (Can't think of a charity where my thousand bucks could actually make a difference, and about which I'm reasonably sure that it isn't really a scam!)

Probably were from a coin counter that rejected them into a tray and they were never given back to the customer, causing the bank to now be in possesion of them, and creating an imbalance in their teller drawer.

I'd wake up from this dream. Feed the cats. eat breakfast, bm, fix my hair (still vain) then off to work.