Will a Shimano MTB 9 speed derailleur work with 10 speed cassette? In general, what issues am I likely to run into, if I use a 9 speed derailleur with 10 speed cassette?  Is the chain width main problem?

I'm interested in Shimano XT CS-M771 cassette/XT M772 Shadow 9 Speed Rear Mech combo.  Will it work?

If it will, what flatcar shifter would work the best (indexed)?
With Shimano MTB components the short answer is No, it won't work, the 10 speed shifters and derailleurs have different pull ratios. 

The long answer is it can be made to work. You just need to define what you mean by work. When I buy XT, I want and expect it to 'work, just like XT should' - reliably, for the life of the components. Its silly to spend XT kind of money to get Tourney performance. Mixing components from different speeds, even when using 'problem solvers' often leaves a less than ideal version of 'works'. 

That said, a few people seem to have magic hands and/or the patience of saints and get these frankeinsetups working smoothly and reliably.