if your accident is severe enough and the other person is underinsured - then you could collect from their insurance (must be their policy limits) and your insurance under the "underinsured coverage" or UIM - Underinsured motorist

yes

Yes, that is the primary function of underinsured motorists coverage.

third party liability coverage is personal liability coverage that protects the customer from damages they incur due to the wrongful acts of others when the liable person is uninsured or underinsured. The coverage is designed to cover bodily injury, property damage and personal injury resulting from occurrences not involving an auto in which a liable third party is at fault, but is uninsured or underinsured. For example, this would cover a policyholder if a dog attacks his child and the dog owner is not insured or is underinsured, or if a policyholder is injured at a party on a deck that collapses and the homeowner is not insured or is underinsured.

Underinsured motorist coverage is a coverage in case you have an accident with someone who doesn't have insurance or enough insurance. For example, if someone hits you and they don't have enough coverage to fix your car, your insurance would help you out.