I can't seem to find the tiebreakers for the standings.  I know that we don't play the extra game to determine the division winner.  I'm pretty sure there's a formula, but I don't know what it is.

Head to head, then division, then run differential, then coin flip.

Yeah, that's the way I thought it went, but I've been in a tie with my division mate many times this season in Shoeless Joe, and although I'm ahead in the season series, and have a better division record and run differential, I'm always listed behind him in the standings when we're tied.  Would it be rearranged on the final day if we are still tied then?

Yes.

I'm guessing that your team's ID is higher than the other team's, that's why you're listed behind him when tied during the season.

Does run differential in the tiebreaker formula refer to who has scored more runs between the teams, or the biggest difference between runs scored and runs allowed between the teams? (and what happens if that is negative for both)

I think it's the difference between runs scored/allowed per team, and as for if they are both negative, it would be the one with the less bad differential that wins it.  I have no reference material for this.  This is what my gut tells me.