various questions on property insurance and tenants

[Steve Chenoweth]

I don’t know much at all about insurance for investment properties. So I feel I should break this down into a series of questions.

Can the quality of insurance you get for an investment house depend on the tenant’s background information (criminal record, credit report, history with property damage, etc.)?

Does house insurance normally have a high or low deductible? Is that dependent at all on the tenant and their history of renting? If so, can I legally run a background check on the tenant?

Are their instances where you can choose to add deductible in the insurance policy if you find out more information about the tenant (criminal background, history with damaging property, or lack thereof, etc.)?

In the lease, can you reserve the right to sue the tenant for any loss it seems that they have caused, like fire or other damage, something that goes past the amount of the security deposit?

Any information on these questions would be greatly appreciated. Thanks!