The answer to all three questions is yes and certainly is classical. 

One simple example is the following:

Let $C_2$ act  faithfully on the set $\{1,2,3,4\}$ in two ways. In the first the non-trivial element of $C_2$ swaps 1,2 and also swaps 3,4. In the second the non-trivial element swaps 1,2 and fixes 3,4. 

Each action defines a representation of $C_2$ on $\mathbb{Z}^4$ via permuation matrices. In one case the trace of the non-trivial permutation matrix is $0$ in the other $2$ so the images cannot be conjugate in $GL_4(\mathbb{Z})$ or $GL_4(\mathbb{R})$ however they both generate a subgroup $C_2$ in the former. 

It is fairly clear this idea generalises to any isomorphism class of finite groups.