1. the company that you hired must be licensed contractor, therefore they must have insurance.
It is the company responsibility to mitigated the affected areas as per IICRC S500 guide line.
The insurance details should be listed on the contract.
You always contact your contractor state license- they can guide you and help you more then you think.
2. You can (but not advised) use your insurance company for the claim, the water loss was accidental and should be covered.
3. You should contact your insurance and let them know about the issue, you don't have to activate the claim and your history record will stay clean.
* from experience, most companies will want you to use your home owner insurance and they will pay you the deductible and the estimated new premium. it is the fastest way to fix your property if the company doesn't want to "damage" their insurance. This move is 100% your call