At used car lots, will some be able to finance without money down and just trade in my old crappy car?

How is your credit? If it's really good, I don't see why not.
But when you go to negotiate, do not mention a trade-in until they start giving you a better deal. Tell them at the end and you will get an even better price.

Ask to look at their Trade-ins.
They know that if they have to scrap them it will cost them money to do so.
So it gives you a little leverage if you can find something you like that you can afford that you're happy to drive away.

Depending on your credit. I wouldn't recommend it though.