You will need to a decent hardware, if you want to run a webserver. I am not talking a high-end server which will cost several hundred dollars. I used to run a web server before, on an OK hardware, 1.66Ghz cpu (non-threaded), 7.5K 250 GB Hard drive and 4 gigs of ram, use to get good performance out of the box, but the only limiting factor will be your ADSL upload speeds, which won't be very fast as your normal download speeds.
So something to think about, if you plan on hosting heavy contents like (pictures or videos).
Initially I bought a domain and purchased a Static IP address. Was paying too much of money for the domain registration, for the static ip address so I cancelled them. If I were you, I would do what Digip suggested, keep your current IP address and go with a Dyndns account, that's what I have at the moment.
For the OS I would probably use Ubuntu, its a very stable OS but you definitely need a decent hardware to run it on.