Lifting & Other Fun Stuff (Questions, help me!)

Absidey

I have some questions. I want to do the NROLFW program, but I'm already enrolled in a class at my gym for a couple more months. I need some help figuring out what exactly I'm already doing, and maybe get some advice about adding more weights.

My class meets 3 times a week:
Tuesday, 7pm, Cardio Night: We do incline intervals on the treadmills + squats and lunges (without added weight) or stations (lunges, pushing a weighted box, plank, throwing a heavy ball, swiss ball core stuff, etc.) Some stuff doesn't seem strictly cardio the way, say, just running or zumba, so I'm a little confused what this is considered.

Thursday, 7pm, Weight Night: We use free motion machines to do presses and curls in a circuit. Planks, v-up, burpees are also in the circuit - things I expect on cardio night. Again - confused.

Saturday, 10am, Kill Me Now Morning: The other nights are intense, then break, intense, break, etc. Saturday is gogogo for an hour, with little quick water breaks. We run to warm up, then lunges, squats, wall sits, planks (with exciting variations), burpees, super fun time with medicine balls... It's like evil gym class.

The questions:
How should I be logging this stuff? Right now I put everything in as circuit training.
Is anything with resistance edging out of cardio and into another category?
If so, does that other category overlap with heavy lifting enough that I should be wary of adding weights?

The advice:
I want to get in at least one day a week of heavy lifting, but with the way my class mixes things up, I'm concerned about overworking my muscles. What I'm thinking is weights Sunday night (36ish hours after my class Saturday, full 2 days before my class Tuesday.)
If the Sunday thing seems to go well, maybe going in early Tuesday for weights before "Cardio Night." This one I'm less sure about.
And keeping pushing to go heavier on Weight Night.

Bonus question:
When I log my weight for lifting, do I count total weight or weight per hand with dumbbells?