Xbox controller battery / battery questions?

It was the first time I bought a new Xbox controller (original). What is included seems to be batteries, not accumulators. I also have an (off-brand) USB C cable and a wireless adapter.

Question 1 - would it be bad to put the batteries in and then use the controller on the cable?

Question 2 - does it then draw its power from the batteries or the cable? I would like to use it mostly on the cable, but also wirelessly every now and then.

Question 3 - can I just buy normal rechargeable batteries and use them instead of batteries? Do I have to pay attention to anything when buying?

Question 4 - will the batteries be charged when I connect the controller to the cable? Or do I have to take them out and charge them in a separate charging station?

I look forward to answers!

1. This should actually work.

2. I mean it takes the power over the cable.

To be sure with both, play only with cables and if you want to play wirelessly then put in the batteries.

3. You can also buy normal AA batteries, but you will need a charger for it.

4. If you have normal batteries, do not charge them via the cable. However, there are battery packs which are also charged during use.

I recommend the original from Microsoft here.