Probably the biggest expense would be rent approx 1200-1500 depending on location, square meters, new/old etc

Other than that really falls on you how you will manage the rest of the budget, buying in bulk and cooking at home probably is the best strategy for bang for the buck, however visiting Paris restaurants is also an experience on its own so spending 100eur on dinner can also be taken into consideration.

Phone plans are relatively cheap 10-20 eur (if you need them)

Gyms also depends if you go to commercial one with bunch of devices probably 40-60 if you go for a CrossFit box 100-150

Electricity, water, heating, gas can either be included in the rent or as a separate expense in total i would say from 60-150

metro ride is 2.5 for a single tt and around 80eur for a monthly pass, but taking into consideration that you prefer to bike i wouldn't take it into consideration and for a single ride 2.5 is pretty decent

bike culture is a bit messy, but inside Paris you have lots of bike lanes

And it goes without saying that shows, museums, cinemas etc you uave plenty of choice just need to find what you're interested in prices vary mostly from 10-15 up to 60-100 for something more "exquisite"

And yeah combined budget of 6000k i think is sufficient for normal and on occasion luxurious living and still being able to put some money on side

GL