After charging the battery for the first time and a quick flight. Battery had to cool down before it would charge again. Anyone?

This is the way batteries should be treated as they will warm up during charging. Othervise they will be completely overheated and can be damaged or can cause fire.

I had the same after the firmware update.   Had to cool battery before it would charge.  

That's normal and this is why I always buy extra batteries.  

Hello there hurstja. Good day and thank you for reaching out. Please do not charge your DJI Mini 3 Pro's intelligent flight battery immediately after flight. The battery temperature may be too high and may cause serious damage to the battery. Allow the battery to cool down to close to room temperature before charging. Charge the battery at a temperature range of 5° to 40° C ( 41° to 104° F ). The ideal charging temperature range is 22° to 28° C ( 72° to 82° F ). Charging at the ideal temperature range can prolong your DJI Mini 3 Pro's intelligent flight battery life. Thank you.

Isn't that how the battery for every DJI drone works and always has worked?  Its how all Lithium based cells work.

Nevertheless, I observed a very quick (over)heating when flying and filming.

I put mine in the fridge for a short while after the firmware update. When I land my battery seems cool enough. The props provide the cooling

I have a problem on my drone mini 2 after the firmware update. The battery was overheating and I lost the connection with RC and drone reviews to switch off?

My battery charges when it's hot, but slower. I have a power meter. Towards the end of the charge, it's slower, so it cools down.
If the battery charge rate is above 24W, it will tend to heat the battery up. Generally, I wait for a battery to cool down, before charging.

I think DJI's guidance is as follows (paraphrased)

Not charging the battery straight after flight, i let the battery cool whilst using another one.
Not charging a battery straight after a FW update, as above.
Not using a battery straight from a charge, as above.

My own preference;
Not forcing the battery to cool down faster by using a fridge, freezer, AC etc as you run the risk of condensation build-up, although, I would use a normal air fan if it was needed.
If anyone MUST use a fridge/freezer, use an air-tight Ziploc bag, although, this is not me recommending the whole idea, i would advise you didn't, but thats your choice

Turn you capture down to 1080p that helps

This is exactly what the instructions advise.