Can I go travel to Puerto Rico with my Temporary Protected Status and enter back into the US?

PR is within the US territory. You are not leaving the US when coming to PR.

Does having a denied immigration affect in any ways in future immigration petitions?

It depends on the reason for the denial.

I'm from the US And fiance is Ghana. We want to get marry in the US

You will need to apply for a K1 visa, I guess. Pictures are good, if they establish the date of the meeting.

Green Card Through My Sibling Who is a Naturalized US Citizen (Country Nepal)

No work permit, no authorization to stay while pending. The process takes about 12 years.

About Immigration

After you file I 130 for her, she might not be able to enter on B1 visa.

If you filed I 485 based on marriage or other basis allowing for an employment authorization, you can file I 765 requesting an EAD based on AOS.

I'm wondering what I can do to help my boyfriend/fiancé of 6 years get out of deportation? Can getting married help him?

You need to hire an attorney to work on this ASAP. You also need to get married to be able to help him.

How soon can I get another visitors visa to the U.S. if i'm currently in the U.S. on the once extended visitors visa?

Sometimes it is possible. You are likely to be questioned why you need to be in the US for so long, if you have a job here, etc.

Our lawyer made a mistake in the O1A petition

Well, I am afraid it might be too late. You should speak with an attorney.

Being a member of denomination as special requirement in I-140 petition

You mean if qualification for an employment listed as being a member of the denomination would be considered legal? The organization's lawyer...