There are 2 champs for softball just like there is for every SEC sport not named football, a regular season champ & Tourney champ. It's been that way for a very long time. Both count although tournament champs are the only ones to get NCAA tourney auto bids.
See above. On the other end of you example spectrum the AU 1999 men's basketball regular season SEC champs had SEC Champion shirts made and I believe there is a banner in the arena. Not sure about rings though.
It's correct. aTm should win the regular season title based on their schedule. If they don't win the tourney they are still regular season champs. Nothing can take that away besides obvious major rule infractions.