Will a right hand shifter described as shimano 10 speed triple work with a SRAM 9 cassette and a shimano compact chainset?
Depending on the derailleur, you may be able to get 10 speed shifters to work if you also swap the cassette to 10 speed. The derailleur may have enough range to sweep the whole cassette. I've had luck using a deore lx 9-speed derailleur with 10-speed shifters and cassette. Worst case for the rear is that you'll need a new rear derailleur.
While Shimano does have one shifter (2010 ST-5603 105) which will work with either triple or double drive trains, no 10 speed shifter works with a 9 speed cassette, at all.
There are cam adjusters which claim to correct cable pull and allow you to use mismatched equipment, but how well they work is open to debate.
You should maintain as close to matching specs as possible. All 10 speed or all 9 speed equipment is pretty much mandatory.
Mixing brands is likely to degrade performance to some degree, although it will work, usually.
Bottom line is, if you need to ask the question, you don't have the necessary experience to make sure that everything comes out ok at the end.
On the other hand, you'll learn a lot if you try it, and if you can afford to experiment, it could be fun to try to make it happen. Be prepared to buy everything 2 or 3 times, if you do try it.
Hope that helps.