Since the car insurance companies are involved it is highly likely that it will be listed on your CLUE report and impact your insurance premium. You should contact your insurance company and let them handle the claim since it will likely be reported either way.
No, you would need to file the claim with the company you had at the time of the incident. They are the ones that would be responsible to pay for the damages since it occurred while you were insured with them.
If you were in an accident, you need to report the incident to your insurance company as soon as possible. Most companies require you to notify them of any incident regardless if a claim is filed or not.
If your husband does not have collision coverage, then he does not have coverage for an at-fault accident. Typically single vehicle accidents (including swerving to avoid a deer) is considered at fault.
Your company's insurance company can find out about the incident if you file a claim. Most insurance companies report claims to CLUE, which is a database that holds claims and insurance information to be shared amongst insurance companies.