Do you feel good during regular daily movement? Do you perform well in the gym? Do you have more joint pain after training than before? If any answer is no, change something. Otherwise, you're probably on track.
This could mean anything. "Dynamic" as in plyo/medicine ball drills? Or "dynamic" as in just moving around.
In general, warm-ups should progress from very general (get blood flowing all over) to very specific (target bodyparts to be trained that day and/or address any individual problem areas like bad back, shoulders, etc.)
I'd say cooldowns are over-rated, but I don't really think coaches go out of their way to recommend specific cooldowns so they're not really rated much in the first place.
For me, a day's general warm-up consists of the 5-minute walk from my house to the gym, a few reps of unweighted reverse lunges with arms overhead, a few reps of plank variations that use arm and leg movement, and the cat/camel drill. And then warm-up sets before most individual exercises.
On occasional upper body days, I'll add band rows and band chest presses each for 1-2x20-30. On days when I squat, I add in old fashioned one-leg hip circles, clockwise and counter-clockwise each side. Like these, but not usually oceanfront or treetop:
My cooldown is the 5-minute walk from the gym back home, that's it.