You may only need a 6-day Swiss Pass for each adult, and a free Family Card for the child. That will cover all your train travel in Switzerland July 1-6 (see below about June 30, and your Lugano plans), plus discounts on mountain lifts. The child rides free everywhere, even on mountain lifts.
It will also cover the full Rigi loop from Luzern, which is a wonderful excursion to a scenic mountaintop by boat, cogwheel train, cablecar, and another boat back to Luzern. The standard route is a boat on the lake from Luzern to Vitznau, cogwheel train from Vitznau to the mountaintop at Rigi Kulm ( opportunities for easy hikes and lunch), train back down halfway to Rigi Kaltbad ( more opportunities for easy walks, and a playground for kids), a descent by c Locarno to the lake at Weggis, and another boat ride (or bus if you prefer) bake to Luzern.
https://www.rigi.ch/en
https://www.rigi.ch/en/experience/rigi-mountain-railways/1-mountain-9-railways
This would be a much better use of a daytrip from Luzern than a trip to Zurich.
As for your other daytrip, I don’t know where you got the idea that it is a long journey from Interlaken Ost to Luzern; there are direct trains hourly that make the journey in under two hours. Look at the Swiss rail map to see all your routes and connections:
https://cdn-doc.myswissalps.com/docs/default-source/rail-network-maps/swisstravelpass.pdf?sfvrsn=7788603a_56
Note that Interlaken itself is not the best choice for a daytrip to Luzern, you need to go a bit further, like 25 minutes on a separate train, into Lauterbrunnen to see the famous peaks and waterfalls of the Berner Oberland. The Swiss Pass will cover that extra distance, as well as further travel up to Mürren or Wengen, the car-free mountain villages, but not beyond.
As for June 30, when you travel from Paris to Lausanne, you can buy a discounted ticket on SBB.ch for the direct train departing Paris at 11:56 for 109 CHF, instead of the regular fare of 155 CHF, if you do it soon. There is no further reduction for having a Swiss Pass on these reduced-price tickets, so there is no point in including June 30 on you Swiss Pass if you do this.
If you choose to wait and pay full fare, then a Swiss Pass will reduce the price on that ticket to 124 CHF instead of 109, but you would have to buy the 8-day Swiss Pass instead.
Check the prices online for the various lengths for the consecutive-day pass and then decide. also consider whether you might like to do more daytrips within Switzerland from Lugano, as that would definitely weigh in favor of the 8-day pass (which is actually a 10-day pass if you buy during the sale).