Where do I start with re-installing OSX on my 2008 MacBook?
I have a copy of Leopard that came with the MacBook. However it fails
checksums.


You can try to repair the disc using this method. Never done it but the author claims he has, and it worked for him. Your mileage may vary.


I have an upgrade copy of Snow Leopard on DVD. (Unsurprisingly?) the
MacBook won't install from it.


You seem to have bad luck with physical media. From this source (and Diago below), the Snow Leopard upgrade disc contains a full installer, allowing the installation even over a system without any OS whatsoever. If it fails to boot, it could likely be damaged.


I have a full copy of Snow Leopard for Mac Mini - I tried to install
and it says it doesn't work on MacBooks.


That is also correct. Minis (like many of the machines Apple manufacturers) come with their own specific versions of OS X. The Mac minis, like the MacBook Airs ship with special versions that differ from the ones you purchase through other channels. Therefore, you cannot use an installation disc from a MacBook Pro on a MacBook or iMac.


I have a new MacBook Air with Lion preinstalled.


Apple machines that ship with Lion cannot be transferred to other devices. If you purchases Lion through the Mac App Store, you could install it on any Mac (except those that ship with them of course).

In all honesty, the best advice I could give you is to book an appointment at one of your local Apple Stores and speak to a Genius. Tell them your situation (bring in the machine and the Tiger installation disc). They'll either give you a Snow Leopard installation for free or boost you up to Lion. I'm not sure if they will give you a free USB Stick, but they'll do something for you. I guarantee that.