Using a router compatability?

Yes, as long as you're talking about replacing the router ONLY and leaving the ISP-provided modem in place.

Most ISPs provide customers with a DSL or cable modem, which can connect a single computer to their network. These modems are compatible with the assorted 3rd-party routers available to consumers (Linksys, Netgear, Belkin etc) You plug the modem into the router instead of directly into your computer, and the computer(s) are connected to the router.

However, some ISPs don't provide standalone modems to their customers. Those customers instead have Gateways, which are a combination router/modem in a single device. Gateways cannot simply be replaced by a router- you'd need a modem from your ISP.

While cable modems are universal (you could just buy one) DSL modems must be programmed/configured by the ISP in order to be recognized on their network, and only specific brands/models are compatible- this is always handled by the ISP before delivering the DSL modem to customers. So if you move crosstown or switch ISPs, the old DSL modem may not work because the equipment it must communicate with at the ISP's end may be different.

If your modem remains unchanged, switching routers is no problem.

the answer is yes. A router is the same no matter what provider you use.

You just have to connect the output of the cable modem to the input of the router and it will work automatically. Wireless requires setup with security.

Sure you can, just tell your new ISP to get you online thats all, any router can be connected to any ISP, thats for what they are made for.