As for the riding issue, wanting him to be more responsive. Simply ask him more things more often.
Spend your practice rides continually changing gaits, changing directions, do serpentines, circles, figure 8s, weave cones, do barrels, do half circles, tear drop change of directions. Everything, but don't stop changing. The more you ask of him the more he'll need to pay attention - the more he's paying attention, the more less you have to do to get his attention back. Always start with the quietest cue and build up. Ask-Tell-Command (Squeeze, Kick, Whack) Repeat this each time you ask him and soon he'll be doing it with just the Ask. Keep him changing, not just gaits, but doing regular, extended and collected versions of each gait. The more you do this the more responsive he'll become. Remember when circling and all to use all your cues, start with your eyes, shoulders, hips, seat, legs, reins last. Soon your horse will be following your focal cues, going where you look, doing what you want the moment you think of it :)
As for the ill-fitting tack. I'd ask the BO or owner of the horse, just say "oh hey, I noticed his ____ isn't working right, do you know what we need to do to fix it?" Make it a question and make it a "we" not "you need to fix this", even though they do. What causes the saddle to slide back? Is he sway back? Back lifts and riding correctly will help fix that, you could also look into using a riser saddle pad, those are cheaper than a new saddle :P or whatever type of pad to even out his back.
Good luck :) Have fun with your new lease!
P.S.: Welcome to the forum! We love pics here!