Chain hops off cassette sprockets when I stand-pedal When I pedal while standing, the chain hops off to a smaller cog. This happens when I "tilt" the bike, as we do when stand-pedal. I am not really sure why this happens, but I think the chain is loose. Is the solution to get a new chain, or is there something I can do?

I lean against getting a new chain, since my current one is ugly and rusty (I have not maintained it very well over the winter!)
You've got a 'stretched' chain that needs to be replaced.

If the chain has been in this condition for some time, and I suspect it has, I always recommend replacing the cassette at the same time. The reason being that a worn chain wears down the cassette and a new, unworn chain, will fit the old worn cassette poorly and the skipping will continue. I had to learn this the hard way, with multiple trips to the bike shop.

As usual, Sheldon has already laid the issue out in depth:
http://www.sheldonbrown.com/chains.html#wear