What are the disadvantages (if any) of accessing the internet through two or more "chained" routers?

The amount of lag is too small to notice. I have 3 routers chained together in my house and don't have any problems. Your DD-WRT firewall will still function and protect you.

The only thing you didn't mention explicitly was whether or not you disabled the DHCP on the modem/router. I would think you want your DD-WRT to handle the managment of IP addresses. Either way, you'll need to turn off one of the two so you don't get conflicts.

There is nothing wrong with this setup. As a matter of fact using two routers in this way is a good way to protect your LAN from a potentially more dangerous network. Your bandwidth will not be affected in any noticeable way.

Sounds reasonable to me. The firewall should work by blocking packets somehow, and having another hop outside the firewall should do nothing. I wouldn't think that adding another very short hop via Ethernet would affect ping time or bandwidth.

The potential downsides as I see them are that you are removing their firewall and their wireless, and substituting your own. It is conceivable that their firewall is better and/or their wireless more secure. Unless you're substituting WEP for their WPA or something, I doubt there will be a problem.