Anyone normal modern car should have no problem being idle for 1 month without starting. Some works fine even after 2 months. It depends on the car's battery health. And also if you have parasitic draw. If your car is still standard without mods like entertainment system or dashcam (connected to fuse) then it should be fine to start monthly. Or if you want to be safe, every fortnight will do.

In my personal opinion, it is better to leave the car idle as it will go into deepsleep after inactivity for long time. In this mode, battery consumption is very low. If you start it everyday, you will need to compensate back the battery because it suck a lot of power when you crank the car.

Do you need leave charger  there even the car warm up for some time?
Can i upgrade my battery but use century brand? If yes which model number?

You all mean this?

Will plan to buy soon, how long can it last  for this gadget?