Properly installed, all air con pumps should self prime. Besides having the pump below waterline, the key is not too have any low/high point in the line between the seacock, strainer and pump inlet. unfortunately, some installers incl. builders often route hose over sringers creating hard to prime system.
No water flow can not hurt your air con as the high pressure switch will trip protecting the compressor. On some systems you may need to reset the high press. switch on the freon line near the compressor. Systems with electronic controls will show a high pressure error and usually self reset.
on older controls, the start position turn the blower on and I believe the pump. Run turns the compressor on. so on Start, the compressor is not even turned on, no damage can occur.
as to running dry, it's never good even for the pump which is why upgrading to digital controls is nice... If flow is interupted, the compressor shuts down and the control will turn the pump off.
check the strainer and make sure you dont' have something stuck in the thru hull. Open the sea cock with cover open to see if you have enough water coming in. note that if you have a plastic bag stuck in the TH, you may have flow in with the pump off but then under suction, the bag is sucked back tight. I always go for a swim to check it out before deciding to replace the pump.
on the march, you can open the pump end and check the impeller, usually without even removing the motor off the its mount, it's just 5 or 6 bolts on the pump housing comver. it's a hard plastic impeller and i've had TWO fail in the past year. i guess with age (7 or 8 years) the plastic got brittle.