Is 'regardless that' the short form of 'regardless of the fact that'?

And is it the same as 'despite that' in meaning?

If it be so, can I write this way?

- Regardless that he is a bad person, she keeps company with him.

- Regardless that it is going to rain soon, he goes out for a walk.

- Regardless that tomorrow is going to be Monday, he won't go to work anyhow.

How do all my sentence above sound?