iPhone 12 sometimes charges wirelessly and wired, sometimes not. Or will start charging after a few minutes??

It should charge almost as soon as you connect the cable or put it onto the Qi-charging station.

Are you using Apple certified accessory, for example a MagSafe?

Is your device updated to the latest iOS version 15.4.1 ?

To rule out a software issue, set your phone back to factory settings, using your computer.

Then set it up manually, without using the backup see if you can charge the phone.

Restore your iPhone, iPad, or iPod to factory settings - Apple Support

If it works again, restore from the latest backup to get all your data and settings back.

Yes, dropping the phone might have caused internal damage to the hardware.

Only a diagnose by an Authorised Apple Service Provider will show what exactly has been damaged.

Use this link to find one next to your place:

Apple - Find Locations