Hi blazej and welcome to the forum, I don’t think that will be the original BMW battery, you can check the age by looking on top of the negative terminal it will have 2 sets of numbers stamped into it, manufacture month and year, eg 01/19 January 2019, the low charge could be because of short trips, sitting admiring the inside and listening to music with the door open and the engine off
, I’d definitely put it on charge for 24hrs and see if that helps, but if the battery is over 5 years old I’d be changing it, it should be an AMG type.
For the cost you’re better off fitting new salmon relays and change them every 5 years or so.
Tyres are 225/35 front and 255/30 rear, I’d stick with that and probably put non run flats on, buy a can of tyre weld and a tyre inflator.
I’m afraid I can’t help with the service reset.
It would also be nice to see a photo of your new purchase