As for the first question, that is what a reduction does. For how to reduce #3SAT to #Monotone-2SAT, see the proof of #P-completeness of #Monotone-2SAT [Val79b], which is based on the #P-completeness of Permanent [Val79a]. To reduce #SAT to #3SAT, Cook’s reduction from any problem in NP to 3SAT is parsimonious and therefore reduces #SAT to #3SAT.
The answer to the second question is no. The reduction in [Val79a] from #3SAT to Permanent does not preserve the number of solutions. Moreover, if a reduction from #SAT to #Monotone-2SAT (or Permanent) which preserves the number of solutions were known, the same reduction would reduce the decision version of SAT to the decision version of Monotone-2SAT (or Bipartite Matching), implying P=NP.
[Val79a] Leslie G. Valiant. The complexity of computing the permanent. Theoretical Computer Science, 8(2):189–201, 1979. http://dx.doi.org/10.1016/0304-3975(79)90044-6
[Val79b] Leslie G. Valiant. The complexity of enumeration and reliability problems. SIAM Journal on Computing, 8(3):410–421, Aug. 1979. http://dx.doi.org/10.1137/0208032