The metro is all steep stairs; the buses have space for strollers. Versailles is the worst possible place to go with a 2 year old unless you are only planning on visiting the gardens, but with such a short trip I would just enjoy the gardens in Paris. Luxembourg Gardens is a good area to stay; in April the place is stunning with bulb flowers and there are excellent playgrounds for toddlers. In late afternoon there may be a vendor with rental toy sailboats for the fountain pool; the child would get a kick out of that. April is when the chestnuts are in blossom and the wisteria and Paris is gorgeous with flowers.
There are lots of play grounds for toddlers in all the big parks, but also most tiny pocket parks around town have toddler toys. If you plan on visiting the Louvre you can do that with a stroller (every staircase in the palace and there are many has an elevator - ask a guard if you don't spot it). The Louvre sits at the head of the Tuilleries park which itself has playgrounds and perhaps the boat vendor and room to run and play (and ice cream and snack kiosks etc). So one parent can take the child to the park and the other continue in the museum if that works for you.
There are carrousels all over Paris, in some parks, and even on traffic islands here and there which the child might enjoy. As noted Versailles is the very worst destination -- it is well out of town and the chateau is a miserable scrum; you would have to carry the child pressed in a crowd the whole time; you cannot use a stroller there (except on the grounds and there is no point going that far for another park).
I would not do the Montparnasse Tower if you are doing the Eiffel Tower and definitely get tickets ahead for the Eiffel Tower. I might consider not going to the third floor as the line for the elevator on the second floor is long and that is miserable with a toddler. You will be carrying the child at Ste Chapelle. And you will be carrying the child up the stairs at the Arc du Triomphe. I am not sure there would be any safe place to leave a stroller at either place, but particularly the Arc du Triomphe. Ste Chapelle is inside a police complex so maybe it is safe to leave?
Montmartre would be a nice stroll on a nice day and there are lots of nice playgrounds; the streets are steep but you can use a stroller there.