That clause question

Hello,

"Pay attent to the fact that he met her."

Can we make it as follows?

"Pay attent to that he met her."

"That he met her is not important." is correct?

I feel "the fact" in "the fact that" is unnecessary and wordy, and so want to take it out.

I wonder whether you can use that clause as an object of preposition or verb, as it acts as a noun clause as a subject.

Where can I learn how to use "that clause" correctly?

Thanks a lot