Is it ok to unplug my laptop battery and just rely on the power cord?

It won't hurt the computer as the power supply is filtered.

Also, batteries have built in serial numbers that are used to track you.

Additionally, be sure to plug the battery in monthly to keep it charged.

Some computers refuse to work without a battery and phones I believe require a battery for them to work.

Reason? Just another way to TRACK YOU.

Laptop manufacturers falsely state the battery run time.

They might claim 8 hours but in real life use, you might get 1/2 of that.

Many laptop makers force dim screens just to make the battery last longer.

That is OK of you sit in a dark closet watching videos.

But not in sunlight.

ya. no actually lol its ******* dangerous keep ur battery in. also ur battery still relies on your power cord lol

NEVER power the PC without the battery.

It filters out the dangerous AC ripple and spikes from the charger

Not using the battery won't reduce the heat.  Many computers won't run without the battery installed. 

You can remove the battery, but I doubt if it will help with the temps, additionally if you accidentally disconnect the power supply or have a power cut you risk damaging your files and losing data, so it's not a great idea

if it works without the battery installed, you're good.  some laptops won't work without a battery

I am not aware of any harm that can be done to your laptop to run it without the battery.

But removing the battery will not reduce overheating.  the processor overheats when you are making more of a demand on it that it can produce.  Removing the battery will reduce the demand on the processor, so it is a pointless thing to do.

Yes. Just make sure you're not moving it around, so you don't risk unplugging it.