Can SK636, SK638, and 636V2 be wired directly to 120V power?

Sorry, please do not do that. 

The internal circuit and wiring are designed to handle low voltage DC current, not high voltage AC. There is no circuitry to handle 120V AC inside the main unit.

You need a UL certified emergency light if you want an AC direct connection.

Our lights use an UL approved AC adaptor which output DC and connect to the main unit to charge the battery, therefore the main unit doesn't need to pass UL standards. 

Are the SK636 and SK638 UL listed?
What type of AC adapter does the SK636 and SK638 use?