So I've got this problem where If I join a public/private game with multiple people it lags like hell and I can only communicate with the host, and the host is the only one that can communicate with me. Like I can only see the hosts messages and the only person that can see mine is the host.

But the strange part is that if I'm the host there is no lag and everything works fine. This started around the time I moved into my new house, so I'm thinking its either the new router I'm using or maybe my ISP or I don't know.

Seriously tho, what the mother tits. Has anyone heard of anything like this and/or know how to fix it?

Probably don't have your ports forwarded on your router quite correctly. First thing I'd look at is your router: working for hosting games but not for joining makes the port forwarding on the router the most likely culprit.

I know that SC can have extreme lag (pausing every 2-3 frames) when there's a "nat" problem, eg two players in the same game are using the same ISP that is doing some sort of natting to the outside world, which I believe causes all of the packets that are supposed to go to p3 and p4 go to p3, and then p3 has to forward them on to p4 (for example).

I would guess that perhaps you have a similar thing on your home network: you've set up the port forwarding to send the packets to a laptop or someone elses computer and that computer says "these aren't for me" and forwards them on to your computer. This is a pure guess.