I recently changed the cassette/cogset and chain on my mountain bike (10 cogs, SRAM 2011 or 2012). The chain rings are worn out and when I stand on the pedals the chain slides pretty unpleasantly. To change more than one chain ring would cost me more than to buy a new crankset so I want to go for this option, however, I've got a fairly old unused crankset 3x for 9 cogs (Shimano Deore AFAIR). Will that crankset work with the 10x chain and sram shifter? If not, is a Shimano SLX 3x crankset for 10 cogs compatible with the SRAM shifter?
Chainring to chainring spacing does generally become less over the different speed generations. There is no exact standard for it. As for the actuation of the shifter/FD, I believe in this case how it works is that the 9 and 10 speed triple Shimano and SRAM shifters pull the same amount, and there's a very small difference in the actuation ratio of the FD to go along with the difference in ring spacing.
Usually combinations like you're proposing here work fine, but it's in the area where nobody can really tell you 100% in the abstract that it will. Carefully test that the chain can't drop between rings, but I doubt that will be a problem. If you encounter any problems, what it's likely to be is that the system doesn't have enough total throw, i.e. you can't adjust it so that the chain doesn't rub on either the inside of the FD cage in the lowest gear or the outside in the highest gear. You may be able to either use an alternate FD cable anchor position or bend the FD to overcome this. Using a 9s FD isn't a good solution because then the cage width doesn't match your chain.
The SLX 3x10 crank should work totally fine presuming no snafus with ring size/FD compatibility.