LG window aircon's fan doesn't start it self.

Called Flash Codes. First try turning off everything at the breaker for at least 2 hours and preferably overnight. Turn back on and wait 15 minutes for self test to complete.

https://www.orionair.co.uk/PDF/fujitsu-trouble-shooting.pdf

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Do the indoor and the outdoor fans run when the cooling or heating is on? The compressor? Is there free airflow thru the radiator coils. Indoor and outdoor? Can only guess "fan relay", a fan's work not being sensed. Or a faulty frost/defrost circuit.

bad compressor or capacitor

how do I set it to heat up the room?

www.appliancefactoryparts.com/content/.../118816.pdf - United States
Above is your service and repair manual, with this you will be able to troubleshoot and repair your Aircon.
I suspect, your thermostat may have a faulty sensor?
factoidz.com/troubleshooting-your-air-conditioning-thermostat/

http://www.inspectapedia.com/heat/Thermostats.htm

Below is a DIY AC, troubleshooting how to. http://www.repairclinic.com/Air-Conditioner-Troubleshooting

Very possible your condenser is extreemly dirty and is turning off on a high pressure cutout. suggest you clean out condenser with compressed air. Brushing is not enough. also, check if condenser fan(at the compressor) is working.

You should also check the outside unit to make sure the coil is not clogged with dirt or grass clippings and the fan is working. The reversing valve that switches it from cool to heat could also be a problem. If it isn't shifting it will not switch between heat and cool.

Sounds like it's low on charge, dirty coil. Is this a ductless unit or window?

Sounds like your aircon is stuck on heating mode. Or. The outdoor fan speed is affected by ambient temperature, or low refrigerant. In heating mode the indoor fan will come on for a while to sense temperature and will then wait for the outdoor unit to pump up pressure and build up heat.