No. Unfortunately, they’re not required to have it. You should ask any lawyer that you hire if they have professional liability insurance because that’s what will compensate you if they make a mistake that harms you.
All communications between an attorney and their client are completely confidential. That means the attorney cannot and will not tell anyone about what you discussed without your full consent.
Yes, you can get a copy of your file from your lawyer. Go ahead and ask in writing, that way you’ll have proof that you did that, and ask them to give you a copy. Just give them enough time to put a copy together, but you’re always entitled to that, even if you fire your lawyer and go hire someone else. You are always entitled to a copy of your file.
The most important thing that you can ask a lawyer before hiring them for a malpractice case is how much experience they have handling malpractice cases. Malpractice cases are a specialized area of the law, and unless you hire a lawyer that is experienced in those areas, your rights may not be preserved.