Most signers are right-handed. A few ones are left-handed. Whichever is normal.
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. In most cases, 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.
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.:)
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.