After a quick check in an online search, I think you would be much safer to keep "that", unless an infinitive verb follows. Please don't assume that because it's possible to remove it, it's always OK to remove it.

Yep, you can.

In everyday language either one is perfectly fine, however, I believe that the technically correct way to write it would be with "that".

Of course. "It is said 'that' Paul works here". One way just sounds slightly different than the other, but they both have the same meaning.

same question.. Is ok if I say: "she said we can run" instead of "she said that we can run"