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If you are in a situation where needing to barricade your door to prevent your parent’s entry is a regular occurrence, it is probably time to involve child protective services. You don’t need to give details of your situation to strangers on the internet, but I would highly encourage you to go to a trusted friend or family member’s home and contact the authorities.
Yes. In the US, a gap of about 3/4 inch from a finished floor is normal. It acts as air return when the door is closed. I know in some other countries they put a hole in the wall above the door for this purpose, usually making it look decorative.
Maybe not the solution, but the ideal answer would be a conversation and family therapy
One important contextual item: how old are you?
Less important but still useful: what jurisdiction are you in? Is your father her enabler, absent, elsewhere?
If you can leave and find a place to live, it is better than living with a narcissist who doesn’t respect your privacy.
My parents also would barge into my bedroom whenever they liked, after a big argument, my dad respected my wishes, but no matter how many arguments about privacy we had, my mother never listened. I began to just be naked whenever I was in my room alone, and she stopped. When I wasn’t in my room, she would still enter, but I didn’t keep a diary and my computer was encrypted.