Are Shimano components compatible with each other? Do I have to get a specific chain?
As long as you use a ten speed chain it shouldn't matter which group you take it from. Normally the higher priced ones are bit lighter but the additional price often does by far not match the lower weight. Also the higher priced ones might last not that long due to the lighter material.

Therefore if it's for a bike where you know that it will see some dirt and wear, better get some mid-priced stuff. It might last longer and if doesn't it won't drill as big a hole into your budget.