The ABS is a router, it's just an Apple-fied router. I personally see no need to pay the extra dough, but if you're a real beginner to networking I understand that Apple's stuff does make it pretty simple. Linksys makes good routers, but their setup instructions suck for newbies and they don't cater to Mac users at all.
Apple's Airport Extreme Base Station (AEBS) unfortunately, does not have an ADSL or Cable modem built in and therefore you must purchase, and attach one to the AEBS to connect to the internet via your network. However, the majority of routers available from other vendors include ADSL/Cable modems, and are therefore more versatile. On the other hand, the AEBS is feature packed and is very easy to configure. It also makes setting up a WDS a doddle!
No, they don't. Most routers do not have a built-in modem. You almost always have to have a separate modem, which most people get from their service provider. I think that's more beneficial anyway; if you switch from DSL to cable, for example, you don't have to replace your router, just your modem.