Core gets done every day, I have always done that because of a previous back injury.
Technically you are supposed to engage the core for ALL exercises, pin loaded, free weights, cardio - Should be switched on all the time for the stability. I also use the pin loaded so I don't need to worry about having a spotter as I go by myself and at times when the gym floor is packed! As for the muscles that is why I do mostly compound exercises not isometric - I eat a lot of protein, eat barely any sugar or carbs as it is I actually need to eat more food because I eat jack all and it's shithouse for my metabolisim but I am getting a meal planner next week.
The Cycle classes I do are all hill and interval based, previous experience has had me burning around 860 cal per 45min session depending on the resistance levels I set on the bike. I only do interval training on the treadmill and bike, I get bored otherwise, 2min hard, 1 min medium, 1 min easy, etc... Bike I use levels 6, 4, 2 and treadmill is level 3 inclune with 10kph, 8 kph and 6kph. I ALWAYS do cardio before and after, works better for me and that way I get a proper warmup and cooldown... And yes I ALWAYS stretch too, also use myofascial release after each session (foam rollers).
I don't have the upper body stability for free weights without a spotter due to 3 previous shoulder dislocations.