In short: Yes! ;)

1) Would I be crazy to run a server from my house? Anyone else here do it?
Nope, that works excellent. /me does it with several servers. I even dug some SSH tunnels. ;)

2) Would I need a separate firewall, or could I use a software one running on Linux?
Go for the software firewall. iptables or ipchains does a great job. It's a standard package in Debian, which has a pretty smooth install nowadays.

3) My box at work will soon have a fixed IP address - would that mean I could securely connect to the box at home?
batkins is right. I'd advise you to disallow Telnet though and only use SSH (for more security).

4) There are plenty of sub-100 pound PC's in Loot around the 300Mhz mark - is that a suitable box?
Again, batkins is completely right. I've even got some 486DX's as servers, each of them working as a SAMBA domaincontroller, mailserver, web/ftpserver, fileserver, firewall and proxy.

>(perhaps even run a Shoutcast station).
ouch, shoutcast is not open source so EVIL! hehe, try the free alternative :icecast or even better, GNUMP3D. It has custom playlists, 3 different levels of compression, ogg vorbis and movie support and other very nifty features. I just installed it myself, very painless installation had it up and running in 5 minutes. check a sample here.

I believe it's written by a monk, but I'm not sure.

>One of the main things I'd like to do is have access to my home files while I'm at work - I don't reckon I'd bet getting many hits so a slow machine would be fine.
Easy, you can use ftp for this.

kind regards and good luck,