Adding a switch on a port on a router does not affect the remaining ports.
I have switches cascaded and near the Satellite receiver and hd-dvd player.
That works most of the time. Some DSL and cable routers limit the number of DHCP addresses they'll hand out. A good hint that this might be a problem for you is if your ISP offers tiered service, like 1-5 users = $30, 5-10 users = $50, etc. Especially if the ISP has control of the router and does all the configuration for you.
Many routers can be configured this way, usually with a starting IP address and the number of dhcp addresses to be allotted. This can help keep dhcp clients from running over static ip address clients, but it can also be used to limit devices on the network.
Otherwise, great way to extend a network. My router has two switches cascaded off it it, one upstairs and one in the other room downstairs.
One thing to consider is that unless your router and switches all have gigabit switches, all the traffic from one switch to another will pass through a single 100Mb/s port. Not a problem for everything accessing the internet, since your internet connection speed is unlikely to be high enough to make that a choke point.
But if you put 4 HD dvr's on different switches and start up a whole mess of MRV connections you might overload the switch-to-router or switch-to-switch single point.
Gigabit ports more or less resolves this, but if you're smart about putting stuff that wants to talk to other stuff on the same switch together you can mitigate it.
Good news is that gigabit switches are getting stupidly cheap. $15 can get you 5 ports and 20-25 can get you 8.
Another really slick setup is if you have a spare router that supports dd-wrt firmware (or want to spend $10-20 on one), you can hook a router up in a remote room and set dd-wrt to client mode. Setup like that, you can plug 4 things into the router and it'll wirelessly bridge those back to your main router, even if that one isnt running dd-wrt. I'm doing that in one room where running wires wont work for me. I bought a $20 5GHz router, put dd-wrt on it, and I've got a 300Mb/s wireless connection from the stuff in that room to the main router.