Can homomorphic operations on many different ciphertexts be multiplied by a same constant?
Yes. Additive homomorphic encryption can also be used to obtain multiplication by a scalar. The way that this is done is by repeated addition, in exactly the same way as you do efficient exponentiation.

Recursively, define:


If $a$ equals 1 then $mult(a,Enc_k(m)) = Enc_k(m)$
If $a$ is even then $mult(a,Enc_k(m)) =
   mult\left(\frac{a}{2},Enc_k(m)\right)+mult\left(\frac{a}{2},Enc_k(m)\right)$
If $a$ is odd then $mult(a,Enc_k(m)) =
   mult\left(\frac{a-1}{2},Enc_k(m)\right)+mult\left(\frac{a-1}{2},Enc_k(m)\right)+Enc_k(m)$


Thus, you only need addition.