I have an extra 10-speed chain I would like to use with a 9-speed cassette on my mountain bike. Will this combination work ok?
I fact, a 10 or 11 speed chain will work BETTER than a 9 speed chain on a 9 speed casstte. The reason is the chain and not the casstte is the limiting factor in a drivetrain-usually
The chain is under such incredible stresses. Chains have evolved to be chamfered, to engage with the front(this derailleur is actually under more stress than the rear derailluer. It simply pushes against the chain to move it between chainrings. The rear derailleur is a more advanced mechanism).
So, if you can alaways get the narrowest, strongest chain that will fit on your bike. And that is usually a higher speed chain than what came with one's cassette.
I would also change the casstte too when changing to a new chain. It's just good bike mechanic technique.