AHRAMIN
Your point number two correct and important.
For example, if your APU TR is u/s(MEL'able), then your APU battery is on it's own in starting the APU on the ground(e.g taxiing in). You'll get one, maybe two, start cycles out of a charged up APU battery in this case, before requiring a charge cycle for the APU battery. A charge cycle, in the case of APU TR fault, I think will not be possible without power through the APU TR. Sometimes a reset of the TR allows this - until it fails again. I'm happy to be corrected on this though.
In the air only, the DC ESS will allow you to start the APU; regardless of the APU battery state(emerg elec config).