I have taught both my son and daughter to hold doors open for the person behind them. I taught them both to open the door for someone with full hands.
I taught them to say Thank you when someone does this for them.
No hats indoors. If you sit at the table at home or in a restaurant the hat comes off.
Standing when someone walks in-no, it had not occurred to me.
Offering your seat when on public transit-yes, it is just good manners to offer your seat to someone elderly, clearly pregnant, or just bone tired looking.