Geoff Lawton's permaculture work on 'Greening the Desert" is well worth a watch e.g. https://www.youtube.com/watch?v=yI9wMtTvWps
(plenty more videos around if you search)
2.If the water table is too deep, trees will need artificial irrigation, forever.
3.It seems easy to create a single-point of failure : monoculture of "best-performing" tree species which would succumb easily to specialized insects or fungus.
2. Tree raise the water table under them, by providing shade (so water on the ground doesn't evaporate as quickly and gets a chance to soak into the ground). They also prevent water tables from surfacing (by drawing the water up from their root systems). In other words, a mature forest will manage the water table, rather than being managed by the water table. You get water table problems if you cut down too many trees. Growing more will solve that.
3. this is true. There are lots of ways around it, though, if your goal is to green the desert rather than creating lots of cheap timber quickly.
Pretty big if.