Right- or Left-Handed?
Which dominantly left- or right-handed are you?
"Some people are dominantly left-handed. Should they sign with his/her left hands?"
Even some people, who are dominantly left-handed in writing, can be right-handed in sign language. In this case, those left-handed people in writing may develop right-handed from the beginning of learning sign language. On the other hand, some left-handed people remain to be left-handed in signing. It is also possible, though rarely, that a right-handed writer can be a left-handed signer.
Whichever right- or left-handed you are, remember that you should be consistent with it. If you are right-handed, use your right hand as dominant. If left-handed, use your left hand as dominant. It is not interchangeable. If you are ambidextrous, you should choose one as your dominant hand and stay consistent with it.
"For lefty people, how do they sign properly? Should they do the 'mirror image' way?"
Yes, but keep in mind that a "mirror" method means either a symmetry or reversal. But, watch out for the exceptions, such as directional movements, locatives, directions, etc. You do not want to give the driver a wrong direction by using a mirror or symmetrical movement.:)
"One has permanent paralysis of her arm? How can one sign with one hand? Is it possible?"
Yes, it is workable. Some ASL words are one-handed. Some others are two-handed, in which some of these two-handed words have dominant-passive roles. One can still understand a word without its passive complement. Let me get back to this later to give some examples.
It is common that native signers talk with one (usually dominant) hand when s/he holds a baby or a box in the other arm. Or, even a cast on her/his dominant arm. People are naturally adaptable.