Need help choosing compatible rear derailleur, cassette & chain
Everything there looks like it should work together all right. If you are replacing the drive train components here, you may want to get new cabling for the shifters as well. 

Any 6/7/8 speed chain should be fine on this setup. The cassettes also seem equivalent. 

I'd worry about the condition of the old shifters due to the age, they should be compatible though. However, Alivio rapid fire shifters can be had relatively cheap you need to replace them if anything is wrong with the old shifters.  The original XTR stuff was definitively higher quality when new.