Satisfiability for various branching programs
For Q2:

For Ordered BDDs (OBDD) both satisfiability and counting solutions is polynomial in the size of the OBDD.

For indexed BDD, IBDD p. 16 satisfiability is NP-complete and the equivalence test is coNP-complete even if there are only two layers. 

In general if a variable is read more than one time it is NP-complete if I remember correctly.