iPhone takes a long time to turn on while charging
This makes great sense if your battery is aging in the normal way for a lithium cell. The automatic turn on happens when the voltage level from the battery raises above a set limit and when the battery is new, it won't discharge as deeply or take as long to rebound when you re-apply charging current.

A simple battery exchange would remedy the situation if you care to remedy this. Just knowing it's normal and expected might let you decide to just be more patient and not drain the battery past 15% so you don't have to wait when you plug it in.

The shutting off is another sign of aging - that won't get better and you can't wait that out - so replace the battery or have the phone serviced if you need that part of the operation more stable.