How do I determine how long I can keep my solenoid powered, for safety sake and before ordering it?
If the data sheet states a duty cycle or maximum "On" time, as in the one you linked, then you can assume that if you exceed that specification to device will overheat and may be damaged.

If the data sheet says "continuous duty" or words to that effect, then you should be able to power it with its rated voltage indefinitely.

As with anything, if you violate a thing's ratings, it will probably be damaaged.