Where on the character sheet could someone add a spot for the Gifts? Or any of the other Theros specific rules? Is there a way to do this, or do we just have to keep track of this stuff personally?

Maybe a homebrew feat?

Not a bad idea.

Yeah I'd say homebrew feat(or normal feat if you picked one of those blessings) added with the feat manager on the features tab of the char sheet.

Those supernatural gifts are part of the planned "general features" update which is in progress and should come out early 2021.

For now, homebrew feats or items that you add to sheet manually is the way most people do it.