She will go out of the door.

I think in American English, the sentences would be
She will go out through the door.

and

She will go out the door.

"She will go out of the door" sounds awkward to my ears.

Don't know about the Brits, Tang, but here in the US we almost never use the of

...or through

as much as that sentence can be made to sound good, this works

In British English, "out of the door" is normal/standard, and "out the door" is informal and may be seen as an Americanism.

Speaking as an American... She will go out the door. She went out the door. She will go through the door. She went through the door.

Arnold raced out of the door, and started...

In its time, it was once reported, this was one of the most often-read lines of fiction in the English language: it is the sentence fragment shown in a brief close-up shot of mystery novelist Jessica Fletcher's typewriter in the opening credits of the (US) TV show "Murder, She Wrote" from 1984 to 1991. I dare say Jessica Fletcher was a little old fashioned.

Historical studies have shown that around 1900, "out of" was the more common in US English, but by around 1940 the "of" was losing ground. This change has not really happened in British English.

A common British expression: "You can take the boy out of London, but you can't take London out of the boy". Substitute town/city/region of your choice.

Somehow, going out through the door sounds like something that might be done by Caspar the Friendly Ghost. Still, out through the doorway is a little cumbersome.

Thanks to all of you.