Good summary of potential disk/drive issues and cleaning options: http://www.retrotechnology.com/herbs_stuff/clean_disks.html. Spatula-shaped cotton wool buds are handy for cleaning drives, as there might not be much room inside.
As per the article, http://stardot.org.uk/ is a good resource for all things BBC Micro in general.
Edit: You can connect a BBC Micro (and many other 8-bit micros) to the standard OSCC. It simply doesn't have the plug on the board, you will need an adapter cable.
https://www.retro-kit.co.uk/page.cfm/content/BBC-Micro-PSU-X...
I'd love to have a beeb to play with (I just missed the wave first time around - went from PET at school to PC at work), but they are going for silly prices on ebay. Maybe a garage or junk sale will turn up something!
I also have a box of 5.25" floppies that I want try and read any data off - they've been untouched for decades, is there anything I should/shouldn't do before plugging in the floppy drive and spinning them up?