Reducing #SAT to #MONOTONE-2SAT
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.

References

[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