Will this handle wet spills or just dry debris?

Yes, this is rated for both wet and dry operation. It'll pull up liquid and moisture without damage, making it practical for drink spills or wet floor mats. Just empty the tank promptly after wet use.

How long is the power cord, and will it reach the whole interior?

Cord length not specified in the product data. Measure your vehicle's interior length before purchase to confirm adequate reach from the 12V outlet to your farthest cleaning point.

Can I use this while the engine is off?

Only if your 12V accessory outlet remains powered with the engine off. Most vehicles cut power to the accessory socket when the key is removed, so you'll likely need the key in the on position to run the motor.