Back wheel started moving backwards when pedalling backwards I have a 10 year old cheap 21 speed mountain bike that I have only ever done basic maintenance on (replacing inner tubes, tyres, brake pads, lubricating chain etc). 

I have just noticed that when I pedal backwards (even slowly) the chain goes a bit loose and then starts to drive the wheel backwards. I'm just trying to find out what might be causing this, whether I can fix it myself, and whether it is ok to ride. 

I have never taken anything relating to the drive mechanism apart, and don't have much knowledge about it. I have noticed that the teeth on the cogs are quite worn, but the chain doesn't usually slip off.
It's pretty normal for a bicycle with a freewheel/freehub to turn the wheel backwards a little bit when pedaling backwards a bit (i.e. if you turn the pedals backwards pretty quickly, you get some very slow backwards turning of the wheel) when the bike is in a stand/lifted in the air. You can sometimes get rid of this by servicing/replacing the freehub/freewheel. 

If the wheel is turning backwards reasonably quickly when pedaling backwards, the freehub/freewheel is shot and you need a new one. 

As for the chain becoming loose when backpedaling, there could be several causes:


Freewheel/Freehub is sticky (you'd need to service it)
Stiff chain (make sure all the links are flexing freely)
Grimed up derailleur (clean it. Jockey wheels sometimes cause this, for example)


That being said, if you're backpedalling like a maniac, most bikes will do this even if they work fine. But at slow speed backpedalling, you shouldn't really see this. 

Finally, I see you didn't say you ever replaced the freewheel/cassette and chain or lubricated the chain in your maintenance for a 10 year old bike. Depending on the wear of the freewheel/cassette and chain, you may want to replace one or both. This does need a few tools (chain breaker, freewheel tool if you have a freewheel / cassette tool + chain whip), but they're relatively cheap/a very quick job for most bicycle shops. 

Your bike probably has a freewheel.