Yes, this has a two-piece power adapter.
This pump has a 5/8 inch connection.
The pump will shut off if the water drops below 1/2 inch.
The pump comes with a 30 foot power cord.
The cord is 35 feet.
Yes. Always best to prime the pump with a solvent. My preferred is Draino.
The pump uses an AC adapter with input of 120v AC and an output of 12V AC. 850mA with a usage of 10.2 watts.
I see no reason why you shouldn't be able to do it. Hopefully the lights have LED bulbs and they are not pulling a lot of power. Otherwise you may have to run another wire due to voltage drop. When making the connection, be sure to use jell filled wire nuts. Using regular wire nuts in the elements could create problems down the road.
Probably not. I haven't actually tested it but I doubt it would get higher than 2'.
No, this would not be advised as it could create a fire hazard.