@Pizzagirl: I added my comments to your list.
Warm up on a rocker board - rock my ankles back and forth while keeping balance - Not bad but need more than that as a warm up. need more dynamic motion of arms, shoulders legs, hips.
Leg swings (just holding on to something) - Again, a warm up. Try to do more warm ups that track what you will be doing when you work out.
Medicine ball - standing on those soft discs, maintain balance and lift above my head / then squat - Like it! Compound, functional movement exercise that engages core.
Cheerleaders (carry weights, hop up / over a Bosu and raise arms as I reach the top … gets the heart rate up) - Same reaction as medicine ball.
Leg press (lie on back, angled, push against the weight) - Ask why you are doing these. Can put a lot of stress on lower back and knees. Unless there is a specific training objective that calls for this exercise, there’s a lot of other things you can do, like squats and lunges with dumbbells, that will hit same muscle groups but in much more functional manner. Squats with barbells are a great total body exercise that hit your quads, glutes and hamstrings just like leg presses.
Adductor machine - legs in between, open them out (the resistance is to opening them out) - This is an abductor exercise. Some machines can switch between abductor and adductor.
Resistance band around the legs - side to side walk, then zombie walk (I LOVE this) - This is a fun one. Side to side is also abductor.
Various arm machines - but the moves are all shoulder / arms, not the bicep ones you are talking about -
it’s basically like doing the fly move, but with hands in different positions - Do something for your biceps with dumbbells or TRX.
TRX - various pushup or pullup maneuvers - sometimes facing in, sometimes facing out - I love TRX. Have your trainer also show you how to do assisted pull ups/chin ups using resistance bands on a chin up bar. Can use for shoulders, lats, back, biceps. Have him show you how to do assisted rowing on a Smith Machine bar (the assist comes from the height of the bar, your body angle and how close in front of you you place your legs.)
Stomach crunches on a slightly angled board - Try reverse crunches too. Have him also show you Mountain Climbers. Work a stability ball in on your ab work. Have him show you TRX ab work.
Then active stretching (he stretches my legs various ways) - Good way to end. But also stretch your back, shoulders and arms.