How to clean and disinfect a mattress...?

Short of having it professionally done (which means encasing the mattress and pumping in all kinds of nasty chemicals) a steam cleaning should suffice. If you don't have a steam cleaner, they can be rented, although the home use ones (like Shark) are reasonable and I'm always finding things to use the steam cleaner on, so it might be worth the investment.

Odds are the mattress has neither of the little nasties of which you speak -- dust mites yes, lice and bedbugs no. I don't know where you live, but in most places in the US it is illegal to sell or donate anything that has had bed bug activity, and I'm sure your friend would have mentioned something about her child having lice. If you haven't seen your friend's child scratching everywhere, chances are good there is nothing.

You can get bed bug traps (available on Amazon as well as other places). Lice is another story. However, I suspect you don't need to go that far, and the most you need is to give it a good vacuuming, then spritz it with some Febreze anti-allergen spray to minimize any dust mites.

http://www.amazon.com/s/ref=nb_sb_ss_c_0_12?url=se...

Bed bugs like to stay in the rolled up edge that goes around the mattresses. If they are there, you can see them.

Vacuum it real good.

If there's some hot sun....put it in the sun. Maybe in front of a window.

Let it air out somewhere.