Is there a group of prime order which could fit the CT-Computational Diffie-Hellman assumption?
The usual technique for having a group of prime size $q$ is to work modulo a prime $p$ such that $q$ divides $p-1$. The target group is then the subgroup of $q$-th roots of $1$ in $\mathbb{Z}_p$. To build such a group, first choose $q$, then selects random values $r$ until you find one such that $p = qr+1$ is prime. This is the way it is defined in the DSA standard.

The remaining part is: how to build $H_1$, the hash function which produces elements in the target group ? For that, you first use a hash function which produces values modulo $p$ (e.g. you use a PRNG seeded with the hashed data, and produce bit sequences of the size of $p$ until you find one which is between $0$ and $p-1$); then, you raised that value to the power $(p-1)/q$. The result is necessarily a $q$-th root of 1, and the whole is a "hash function".

As far as I know, such a group would fulfill the CT-CDH assumption -- i.e. there is no known way to break it. CT-CDH is a "weaker" assumption than standard CDH, but there is no proof that it is strictly weaker.

For your additional question: $R$ knows the $a_j$, which are random non-zero integers modulo $q$. $R$ can thus compute each $a_j^{-1}$ modulo $q$ (that's regular modular inversion). In the expression "$(D_j)^{a_j^{-1}}$, $D_j$ is part of a group of size $q$, so any exponent can be taken modulo $q$.