So when you turn it on normally, on battery alone, it won't turn on? Does it come on when the charger cord is plugged in? (not referring to a power bank, just the charging cord).
In regard to the buttons, if they aren't working, then you likely need to have them looked at to see if they need replacing or are just stuck. This would also have an effect on its ability to start up correctly.
If the tablet is under warranty, I would contact the manufacturer about the repair/replacement (including the start up problems). However, if it isn't under warranty, then I would try a local tech. None of your issues seem software related, and if it is hardware you are going to need assistance to resolve it.