Take into account that a lot of the people on the Eurostar will be be lining up to take taxis, so you may have a somewhat long wait if that's your plan. You may be better off taking the metro (or a bus is better) to somewhere closer to St. Michel and then a shorter taxi ride to your hotel. I would not plan to do anything else before the Eiffel Tower. If you are staying close to Place St. Michel, you'll walk to Notre Dame (it's about 7 minutes, maybe 8 if the traffic lights are long).
Arc de Triomphe - there is an elevator. Mom will then have about 2 flights to climb from the gift shop to the top. You generally get there by taking the metro to Charles-deGaulle-Etoile, then walking a long corridor. If you are buying your Paris Museum Pass ("PMP") there, the ticket office is underground just before the stairs up to street-level. With PMP in hand go to the entrance to the Arc and ask to use the elevator. After enjoying the view up top, go back to the gift shop and if there's no one at the elevator, ask where the operator is. The only reason to go to the top is for the view. If you just want to see the Arc (worth it), you don't need the PMP here. After you walk around the Arc, go back down to the tunnel and take the metro to your next sight.
Sunset on June 10 is at about 9.55 p.m. If you want Paris after dark, you'll have to wait until about 10.30. It's stunning, but it's after dinner and maybe after an afternoon nap too. Any time is good for a Seine cruise. Do it twice, once in daylight and once after dark. If you only go once, make it after dark! My recommendation is the Vedettes du Pont Neuf (you can get a discount online in advance).
The Picasso Museum is not included in the PMP and you would have to get tickets in advance (and lock into a timed entry).
I'm not sure you need a 4-day PMP. You only have 3 days to use it, so maybe rearrange your schedule to do those things in 2 days? Besides the Picasso, the Opera is not on the PMP. Entry to Notre Dame is free. The PMP is only to climb the hundreds of steps to the bell tower.
Versailles is a long day. The site is huge and you will want to see the outlying buildings and the gardens as well as the palace. Take advantage of the tram/shuttle (whatever it is) that takes you around the grounds. It is too far to walk. I would not plan anything intensive that evening. Might be a good time to relax on the Seine cruise. On Saturdays, there are fountain shows. If you want to see the fountains, you need to buy tickets in advance. It is likely to be much more crowded on Saturday because of the fountains. You may want to skip them and go earlier in your stay.
Sainte-Chapelle is one of my faves. It's on the PMP, a 5-minute walk from Notre Dame (less from Place St. Michel). To maximize your time, forget the PMP and, instead, buy tickets to an evening concert (they cost €34 or €29 - I prefer the stlightly more expensive ones - but don't waste money on the VIP tickets). Get there before the doors open and have time to walk around and take photos, then relax and enjoy the stained glass windows while listening to the music. Then you save waiting in the usually very long security line during the day. There are two a night, one at 7 p.m. and one at 8.30 p.m. I think it's better at 8.30, as the sun is lowering in the sky and the angle of the light through the stained glass changes. Then you can walk another 5-10 minutes to Pont Neuf for the 10 p.m. Seine cruise beginning at sunset and ending after dark.