Noob question here. I know training a horse raises the training gauge, but does showing affect it too? In other words, does training plus showing 3 times a week, even if it does poorly, boost the horse to the next level faster than training alone?

Showing has no change on the training gauges, but the training gauges effects showing

To be profitable in showing you must train your horses. By training your horses the training bars go up

Horses will level without ever being shown. Shows get you money and horses points for ranking on leaderboards, futurities, federations, etc.

Don't show horses that won't finish "in the money" (7th place or up), because you'll just waste money to get no reward. You'll get no profit and the horse will get no benefit.