Right- or Left-Handed?
Most signers are right-handed. A few ones are left-handed. Whichever is normal.
"I am dominantly left-handed. Should I sign with my left hand?"
If you are dominantly comfortable being left-handed in sign language, then you can sign with your left dominant hand. Though even some people, who are dominantly left-handed in writing, are right-handed in signing.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.
Whichever right- or left-handed you are, remember that you must 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.
"As a lefty person, how do I sign properly? Should I do the 'mirror image' way?"
Yes, but watch out for the exceptions, such as giving a direction, describing a room, and such from a signer's perspective. You do not want to give the driver a wrong direction by using a mirror approach.:)
"I have a permanent paralysis of my arm? How can I 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.
Video above: an ASL word "wedding" signed by seven-year-old, right-handed Juli holding a cat in her arm. Normally, the sign for "wedding" is two-handed.
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. Humans are naturally adaptable.
Learn about dominance rules why it's the way it is.