To answer your questions in no particular order:
For an ADSL service, you must lose the modem (and you would be wise to uninstall all trace of it, but that's another issue) and use an ADSL modem/router.
Such as a 3Com 3CRWDR100A or a Linksys WAG354G - it is wise to saty with established networking brands (3Com was founded in 1979, for example) that have a knowledgeable userbase.
You should not be worrying about (quote) "fast" routers unless you intend to perform cross-network high bit-rate media streams or transfer very large quanitities of files on a very frequent basis.
Adapters? Use Ethernet where practical. PCI wi-fi with an external aerial otherwise.
Routers do not use any form of software or drivers: they do not install or have any direct connection to their cleints - the LAN adapters act as intermediaries.