With your itinerary I think a Half Fare Card would bring you the most value.
Many of your days are far too ambitious. Consult www.sbb.ch to understand how long it takes to get from one place to the next. Also, some of the walks you are planning will likely not yet be accessible in early June.
>Day 02 Do paragliding, do attractions like Schynige platte and Harder Kulm and if time permits do Seil Park<
Paragliding will take several hours. It takes 1.5 hours to get from Lauterbrunnen to Schynige Platte then, of course, another 1.5 hours to get back. It is just under an hour to get to Harder Kulm from Lauterbrunnen, so 2 hours round trip. This is transport time only, no time to enjoy the area once you arrive.
>Day 03 train to Jungfraujoch (do ice palace, and would they have dog sledding in June ??), then train to Grindelwald and do the hike to Bachalpsee lake then return and do the gondola ride to First Bahn. Would we have time to even do Glacier Gorge ????<
Trip to Jungfraujoch from Lauterbrunnen is 1 hour 45 mins each way. Train back down to Grindelwald is 1 hour 45 mins. Walk through Grindelwald to Firstbahn, about 20 - 30 mins if you don't stop along the way. Firstbahn up to First is about 20 mins. Walk to Bachalpsee is 3 hours round trip. However, it is very likely that this walk will not yet be opened due to snow.
Annika's recommendation re: your Mürren, etc day is exactly what I would suggest as well. Any day you want to go up one of the peaks needs to be a clear day AND you need to go in the morning as clouds often come in during the afternoon and will obstruct the views.
The cities in the region pale in comparison to the Alps. I would not spend time driving to visit these cities when you are based in paradise! Brienz is charming but is easily (and more fun) accessible via a boat cruise on Lake Brienz. I recommend dropping the Berne, Thun, Gstaad plan and spending the day on the boat hopping on and off to various quaint villages. Or go look at Brienz and continue on to the Ballenburg Open Air Museum which you can easily reach via bus from Brienz. The staff at the train station in Lauterbrunnen can sell you tickets for the entire outing and help you with the schedules.
Then Day 7 relocate to Lucerne for 2 nights. Lucerne is the loveliest town in the region. It is only about an hour to Zurich airport from Lucerne so you could spend your last night there before flying out of Zurich (?) the next day.
A car is not advisable for this journey. It is all easily do-able via public transport. Many of the places you want to go are not accessible by car anyway.