When I asked a few questions about 9/10 spd compatibility a few months ago, the view then seemed to be that you’re more likely to be able to get SRAM 9spd stuff to work in a 10spd setup than Shimano, but by no means is it guaranteed to work.
The biggest problem you may have though is that 9spd mechs are only designed to work with a maximum rear sprocket of 34T, a 36T rear cog is likely to foul with it. Obviously the 10spd mechs have been designed to work with up to 36T sprocket.
Personally, I’d just buy a 10spd mech and sell your 9 asap in good condition. Save you a few hours of faffing probably, and you know it’ll work!
And as far as chains are concerned, of course you can use any brand on any sprockets, same as you can with 7/8/9 speed. Just make sure you don’t buy a 10spd roadie chain, as the cogs (and the chain) are narrower than on MTBs!
You ridden a bike with 10spd yet? Did you find it at all beneficial?