Yes, most definitely.
Check the state license through their look up database or call the Contractor licensing office. On the insurance, ask them to have their insurance company send you a certificate of insurance or call their insurance agent.
It is important to have a contractor who has insurance: 1) It is not cheap and takes a little while to get, so the contractors who have insurance are more stable, aka responsible. If you have a project that has problems due to contractor error, there is help. 2) By law, any licensed contractor must have insurance as part of doing business.
Yes, don’t let anyone start your project with out Insurance and make sure you are listed as an additional insured on the policy. In most states, a non licensed contractor cannot pull permits. The money saved by using a non licensed contractor in most cases will end up costing more and turn into a nightmare for you.
Extremely important not only to protect the homeowner but it speaks volume to the contractors character.