Yes, I think al of this can be configured through software. I don't have any BT headphones, but I know that, for example, spotify starts playing immediately when I connect the phone to my car's system.
As others have said, check if there is a firmware update for your headset.
Check if the audio app you are using supports control through headset buttons.
Disable any battery optimization for your audio app.
And in the end, consider that Huawei phones are running EMUI, a Huawei-modified Android "skin". Each manufacturer modifies Android in their own way, and the version shipped by Asus is not as modified as the one from Huawei.
And again, maybe try Poweramp? Here you can see some of the BT-related settings it offers: