We certainly have some unusual mascots:
Hutto Hippos
Groesbeck Goats
Amarillo Hgh Golden Sandies
Grandview Zebras
Robston Cotton Pickers
Brazosport Exporters (I've no idea what that actually is,

)
Springtown Porcupines
New Braunfels Unicorns
Winters Blizzards
Farmersville Fighting Farmers
Coleman Bluecats
Hamlin Pied Pipers
Roscoe Plowboys
Mesquite Skeeters
Central Catholic Buttons
Randolph Rohawks
Frisco Trash Pandas, oops... I meant Raccoons

No idea what the most popular mascot is, I'd guess Tigers, Lions, or Bulldogs
Mascots that don't end in S? Hmm... none that I can think of
Live mascots? also, none that I know of? Might be a school or two with a bulldog maybe
It would be UNS for what the coach would do to the mascot and probably an immediate ejection and possibly charges filed

For seeding tiebreakers, a district can choose to use a system called positive points but I've no idea how it works. I do know if the districts has adopted positive points, they are required to inform the white hat because it can potentially alter overtime rules, I think. No districts in this area use it (that I know of). Outside of that, I think they just do a coin toss.