Having had room a/c units for 40 years, I thought I knew how to use them. The indicator that connects to the thermostat may be slightly off, but I like to keep the room at 26. The unit was just recently serviced by professionals, cleaned, freon recharged, etc. It cools well.

I came home, and it was 30. I turned the thermostat to 18 and waited (this is a wall unit, no remote control). When the temp in the room reached 25, I put the thermo at 32. It still kept cycling! Now it's at 34, the temp in the room is 26, and the unit still cycles on and off about every 5 minutes. My partner insists that the off/on cycling of the compressor will surely kill it.

Is this normal, and no cause for concern? Is my boyfriend right that I'll burn out the compressor? And do we all agree that if you want the room to be 25, it won't cool down from 30 any faster, whether you set the thermostat at 11 or 25?

The control has no effect other than setting the turn-on/turn-off temperature. It will not cool faster at a lower setting. But many units do have high fan speeds or other terminology for a faster cooling. This is the hot season so if outside temperature is high, or sun on wall the unit is in, there will not be much time between cycles or you room will heat up beyond your liking. There is often a night or sleep position that will allow a larger temperature spread before it comes back on but I find that very uncomfortable as you either freeze or sweat. New inverter units are likely much better but don't have one myself. They advertise a much tighter temperature control.

it depends where the thermostat is sited.

if it is high on the wall , it may read 30+ degrees , but the temperature at floor level , or at bed level will be lower , possibly 5 or 6 degrees lower much lower.

cold air is heavier than hot air and sinks down.

1) Is this normal, and no cause for concern? Is my boyfriend right that I'll burn out the compressor?

2) And do we all agree that if you want the room to be 25, it won't cool down from 30 any faster, whether you set the thermostat at 11 or 25?

1) it is not normal but it won't "burn out" your compressor. what might burn out if the cycle is a few seconds is the relay that triggers the compressor. remedy is to increase the thermostat's on/off range (a little screw has to be adjusted).

we all agree.

Check the temperature sensor inside the unit and make sure they have not pushed it back into the cold coils when they serviced the unit. If they have the unit will only cool for a minute or two and then turn off thinking it is too cold, only to turn on again as soon as time delay relay allows. The sensor will be a wire like thing, normally on right hand side and just behind the removable filters. Often a plastic covered loop looking thing. It needs to measure the temperature of the return air flow and not the cooling coils.