i had a '99 australian-spec 156, so not exactly like yours...
.. does anyone know where I can download a workshop manual for the car?
ebay. buy a copy. haven't seen anything online, and wouldn't expect to.
what pads and discs does it take and where's the best place to get them (ie cheapest!) Going to change them myself, anything I should know?
first decide what you want: OEM or upgrade. visit any of the major brand-sites and you should find them listed, with specs. when i changed discs, i bought tarox G88 40 groove versions; more for bling than anything! weren't cheap (esp as an import), but looked great. little bit noisier when braking (v slight thrum) but no big deal.
pads: again, OEM or upgrade. ferodo or whatever. i used EBC greenstuff, but they weren't anything special. at least as good as OEM, and much less brake dust.
i never pushed my car to the limits, so never experienced fade, so it suited my driving fine.
changing them yourself is do-able. you may need an 'alfa special tool' to work on the rear brakes (from what i've read, though my memory isn't great!). i've not done it myself, but if you have good tools, know how to use them, and can follow basic principles, you shouldn't have problem.
The computer on the centre of the dash appears to work but only says I've done about 7 miles for every 10 or so I've done. As a result it also thinks I'm doing 20mpg too! Is this common? Should I ignore it or can it be fixed easily?
no idea. depends what sensors it works from and whether they need replacing.
The battery needs its code, I'm waiting to hear back from the garage I bought the car from about it but don't suppose there is an easier way to get it or a default code?
battery code? not heard of this.
There is no spare wheel or changing kit. Just bought a space saver wheel on ebay. Is there somewhere else I can get a cheap standard jack for it?
was this standard for that model or your market? i had a spare tyre with jack. a wreckers or alfa dealer can help no doubt.
It has no owner's manual, where can I get a new one?
probably through your alfa dealer.