What determines viability in grafting a tree,  like a cherry branch into a peach trunk.  Do the classification taxonomy terms tell what is compatible ?,  clade, order, family,  genus, species...

Also, Is it possible to graft a branch of a male say pistachio tree onto a female pistachio trunk and obtain a self pollinating (by proximity) tree species  ?

I can't see why not. There all sorts variations of plants grafted together.