I have had toner explode/spill all over printers all the time, it's a pain! My recommendation is to use a vacuum with a hose. It's probably not a supported method but I've had pretty good success with my shopvac. Also, tissues which are designed to pick up and keep small particulates make the best cleaning rag in my opinion.
I would look up the repair manual online, disassemble the printer (make sure you can put it back together again), vacuum and wipe everything down with tissues. It's a lot cheaper than a new printer or paying for a repair man to come out and replace all the parts that are just a little dirty.
To answer a few of your other questions: A drum is the part of the printer that transforms the image to be printed from the laser to the paper. It's basically a specially coated cylinder that the paper wraps around and receives its image from. Replacing the drum is not always a viable option since in some cases it's not meant to be user replaceable and on a printer as old as yours I couldn’t find any for sale in my 30 seconds of searching.
Good luck.