Hi Rick--Yes, you can transplant your rose now, but it would be better to wait until early spring, after the last frost, when the rose is still dormant. This will cause less stress and shock to the plant. When you move it (or right before you do) cut the canes back to about a foot or so and remove any existing leaves.
A deeper container is better. You can just use any regular potting soil.
Unfortunately, I'm not a good resource for starting roses from cuttings. I've tried several times, following instructions, but the cuttings have never rooted. Try googling "growing roses from cuttings."
Good luck with your rose projects.
