Day in Luzern and Rigi

Hi!
1) It is a perfectly reasonable itinerary to me. Half a day is enough for the highlights of Luzern.

2) maybe start with the Lion Monument (take the bus #1 if needed, it's covered by the pass), and work your way back to the old town

3) not done, but Kulm to Scheidegg looks long!! You will not have time. And Kaltbad is on the wrong line to get back to Zürich.
A very worthwhile walk is the one to Chänzeli viewpoint, accessible from the top of the cable car up from Weggis, or from the Kaltbad stop on the railway up from Vitznau.

4) By the way, boat to Weggis + 10 min walk + cable car is usually faster than boat to Vitznau + cog railway, and no less scenic.

I don’t know if it is of interest to you but Lucerne is home of the Swiss National Transportation Museum. Trains, planes & automobiles. Boats too!

Although it covers the high points, your ambitions for Luzerne city are ... modest. We like art museums, and there's a concert hall too. It is usually helpful to give the date or month of a planned trip, whether in high season or not. Google Maps will tell you if you want to do this entirely on foot, which I consider reasonable. Also locate the ferry dock on Google Maps.

You have to accept that allocating a specific day to a Swiss mountain ascent includes the possibility of a misty summit. There are webcams, but you have a rigid schedule. AFAIR, the cog railway is right beside the Vitznau ferry landing. The cable car is a different side of the mountain.

We did not do any mountain walking, because we were sleeping in Vitznau, and instead, walked back (along the shore and highway) from the cablecar to near the cog railway. This is (physically, not scenically) a road-shoulder walk, mostly without sidewalks, and I'm not saying you should do it. We also found a restaurant (along the road) to have lunch. I forget why we didn't eat up on Rigi. Probably it was crowded or steam-table-y.

Obviously you need to consider mountain temperatures, safe clothing and hiking shoes and gear.

You can get to Luzern by train from the rail station in the Zurich Airport in 1.25 hours (and vice versa for the return). You could skip Zurich altogether. Going up the mountains is weather dependent, and i think the weather in the morning is usually better, but you can always check with your hotel (they have cameras on the mountains). So you might want to go up first thing, on whichever day looks like the best conditions.

We really enjoyed taking a cruise on the lake for a few hours.

I assume they are traveling from Paris to Zurich by train, not flying.

Skipping Zurich and heading straight to Luzern would be a good suggestion if they weren’t going to ride the Bernina Express on day 3. But since they are, and they stayed both nights in Luzern, it would add an hour to their travel time to Chur to board the Bernina Express, and that is already a long travel day. So they would need to go to Zurich at the end of Day 2 to make that a better trip, which means changing hotels.

If they are OK with that, then it works—-the train from Paris to Luzern is only 30 minutes longer than the trip from Paris to Zurich (5 hours versus 4.5 hours, both with a single change at Basel).