Bike throws chain?

[Skyguy9999]

I have another bike (besides the one mentioned in my first thread) which seems to have a habit of throwing the chain off the back gears when riding in any gear (or spontaneously shifting down) - and sometimes into the pedals. The worst was when the front side shifted down while riding and jarred the bike. Then on the very next ride, the chain got derailed off the derailleur pulley itself before I got 100 feet.
The bike has these Falcon 10-click shifters for both front (3 gears) and back (5 gears), so I have to wonder if that's part of my problem - the chain grinds against the sprockets when it's not on the exact spot that is needed (you have to click and ride until it pops into a gear and then "center" it so it doesn't grind), which I'm sure is causing problems.
Does anyone have a suggestion on a good direction to start in dealing with the problem that was described? Or would it require ditching the shifters, which I'm thinking probably would help tremendously anyway. I've had many attempts made at fixing this bike, so I am curious about figuring out what would solve this problem for good.
Edit: I just got done working with the bike and found the chain off the first derailleur pulley three times. So I'm thinking now the derailleur is not putting enough tension on the chain... Could be wrong though?

[Skyguy9999]

I didn't particularly double-check the rear derailleur, since it was (basically) shifting fine. I went through the whole process and found the rear cable very loose with the shifter all the way down. I guess whoever worked on the derailleur beforehand would have had to tighten the cable with the chain in the bottom position, but the shifter not in the same place?
So I tuned the derailleur and rode it about 30-45 mins and didn't have the chain come off the derailleur pulley. Something new learned in the line of troubleshooting I guess?
Anyway, one question still remains. I can see now that the <a HREF="http://bnfbua.bay.livefilestore.com/y1pN0RuMTorzX2k9D-DiFkckxbLQywHxetSfdbl1ddfU9DJixwCbFqHQJ7LXOqJsuv0W5KXb0Nn1tc/falcon%20thumbshifter.jpg">Falcon ML-10 thumb shifter</a> is as annoying to work with as it is to ride with. This means that I will likely want to replace them anyway in due time. Could I just replace those with index shifters (that would have to clamp onto the handlebars), and be fine with the derailleurs that are on there, or do the shifters need to be matched with the derailleurs?