It can be done, preferably by a stylist, but is not the best thing for your hair as it might cause breakage or other damage if your hair is not in the best of shape.
It's not a good idea to color your hair after a perm. It should be the other way around - color first then a perm. You will mess up your perm or lose your perm if you touch it up with color.
It can be done but it is best left to a professional. Mixing the bleaches of the hair dying process with the chemicals of the coloring process can turn out disasterous unless you are trained to do this. If either of the processes were done at home then I would recommend you talk to your stylist to see about it.
Nopes you can't its also a safety precaution on a lot of such products. There should be at least 2-3 weeks or more gap between perming & coloring
for scalp skin's safety purposes.