It's certainly possible to attach a second router in the way you described - you might have to tweak the routers' configurations a bit to make it work properly, but in principle, there's no reason why it couldn't be made to work.

There may be some performance issues doing this, but not because of the Ethernet connection - Ethernet, unlike ADSL, either works or it doesn't, and if it works, then it'll work at full speed. The performance penalty is a consequence of all your data having to be processed and routed through an extra device. It won't slow download speeds, but it might increase "latency" a bit. If you play online action games, you might find the controls a little bit more sluggish, but otherwise, you're unlikely to be able to see a difference.