Yes its perfectly normal. Inside the front part of the unit is a thermometer. As the temperature in the room heats up to 73 (or so) the unit turns on and blows cool air until the thermometer reads an ambient temperature of your chosen 72 degrees. During hotter weather it will stay on longer, cooler less.
It is indeed normal for an AC unit to cycle on and off, regardless of outside temperature. The AC unit will continuously try to remain at the 72 degrees you set it to. If the temperature inside is higher, the unit will cycle on to cool off. Once it's reached the goal, it will cycle off.
If the thermostat is not responsible for turning the unit on and off then no. There are several problems that could cause this. A faulty heat sensor on the compressor or the system being low on refrigerant could be possibilties. Maybe the thermostat itself is faulty. If you do not have knowledge about these things, call someone who does.
My small unit does this too for some reason. Once it has been running so long, it needs a break, then will come back on a few minutes later. I was told this is normal for them to do this. You might clean the filter just to make sure that isnt causing problems.
Air Conditioners cycling on and off is not something to be worried about it. If it is not cooling or heating up then that is a problem, but most Air Conditioners are set to go on and then turn off. If you want it to cycle all the time, then you can set it for that too.