Let $D(M,N)$ be the set of all possible degrees of maps from $M$ to $N$, $M_1\#M_2$ the connected sum of $M_1$ and $M_2$.

1. Can $D(M_1\#M_2,N)$ be calculated in terms of $D(M_1,N)$ and $D(M_2,N)$?
2. Can $D(N,M_1\#M_2)$ be calculated in terms of $D(N,M_1)$ and $D(N,M_2)$?

Here all manifolds are assumed to be of the same dimension $d \ge 3$. I'm especially interested in the case $d=3$.