I use the official Samsung Black Leather back case.
I also use a dbrand camera skin module cover.
And a Whitestone Dome Glass Screen protector @Moomin8
There are cases that incorporate a battery but these can add quite a bit of bulk to the phone.
I easily get a days usage out of my phone with around 50 to 60% battery left but we all use our phone's differently and set them up differently so your usage will differ from mine. It's a subjective topic.
Current Phone ~
Samsung Note 20 Ultra 5G SM-N9860_256Gb SnapDragon 865+
Samsung One Ui 3.1/ Android 11.