Having travelled three times with my little girl, first 10 months, then 22 months, then when three. On our first trip, she did not sleep on the plane at all. So I had to stay awake and keep her quiet and entertained. Hopefully, you will be flying with an European airline, they treat kids much much nicer than US based air lines. Day one with no sleep is brutal. My little girl slept in a child carrier during the morning we landed.
I would suggest visiting your important sites when they first open in the morning. We would take her out of bed and let her keep sleeping. She was more patient and quiet when she woke up. As the day progressed, she would get too noisy and antsy for museums or quiet places, so one of us would take her outside while the other would enjoy the museum.
You mentioned a stroller. We prefer a backpack, Osprey Poco Plus, to a stroller. It is extremely comfortable to carry! We found one at our local REI. No problem with stairs or soft gravel. It was comfortable enough for her to sleep in. We started preparing her many months ahead of time by carrying her around in it. Many places in Paris have soft gravel the small umbrella stroller wheels will not roll over. A stroller with large wheels will roll without a problem. Paris also has lots of cobblestone sidewalks that may be rough to your child.
We used Gelato to bribe our big girl. We stopped at parks everyday for our kids to run around and burn off energy and as a reward. Yes, add a day to Paris!