Any polynomial which is hard to count but easy to decide?
(I am posting my comments as an answer in response to the OP’s request.)

As for Question 1, let fn: {0,1}n→ℕ be a family of functions whose arithmetic circuit requires exponential size.  Then so does fn+1, but fn+1 is easy to decide by a trivial monotone arithmetic circuit.  If you prefer to avoid constants in monotone arithmetic circuits, then let fn: {0,1}n→ℕ be a family of functions such that the arithmetic circuit for fn requires exponential size and fn(0, …, 0)=0, and consider fn+x1+…+xn.