That's right. One airline I worked for carried contingency insurance for some of the situations mentioned above.
I am fairly sure that an airline would not be permitted (by the CAA in the UK) to operate an aircraft without the hull being insured... they absolutely would not be able to operate without liability insurance for passengers, employees, and third parties (which costs more in any case).
As Boss Raptor says, there is no automatic payout by an insurance company to an aircraft lessor in the event of a hull loss. If the insurance company had a case for negligence or wilful damage by the airline (or its employees), they won't pay, and it will be up to the leasing company to sue the airline (which is what I was trying to say above).
The Silkair and Concorde crashes are good examples of what the insurance companies get up to when there is a whiff of culpability on the part of the airline...