Welcome to My Swiss Alps.
That sounds like a generally good itinerary, but you will have some long travel days.
What time does your flight land in Zürich and will you be coming from a Schengen Country ( saves time on passport control lines) ? Might affect the Rhine Falls trip.
June 5th is problematical.
You have to get to the start point and finish point of the GEX, from and to Wengen, unless you only take part of the route.
June 6th is awkward, but doable.
First is above Grindelwald, and Schlthorn is above Lauterbrunnen or Stechelberg.
It takes quite a while to get between those two branches of the valley. Look at the map.
First Flyer - .
Their web site says it is open when you are there:
The cable way to First from near Grindelwald is running on June 6th. The journey to First from Wengen takes between 1 hr 40 minutes and 2 hours.
The best way to figure out feasibility for any journey is to look at a map:
and check the timetable:
Wengen is my favorite village in Switzerland.
However, for extensive travel outside the Bernese Oberland, I prefer to stay in Interlaken. The journey from Wengen out to Interlaken Ost ( IO) for connections to the rest of the country takes 51 minutes, meaning that you waste almost two hours of each day on making that initial outbound + plus final return leg.
It won't affect the travel you have mentioned, but you should note that in early June near the Jungfrau, many facilities are closed. It is between the ski season and the hiking season, but conditions are not good for either one. So, few tourists and little business for restaurants and hotels. June 15th is the npmima date for start of the Summer season, but that date depends a lot of the snowpack left from the Winter.
Most of the mountain transport starts up on about May 24th ( apart from the cograil for Kleine Scheidegg and Jungfaujoch, and the Schilthorn Bahn which are open all year ( JFJ) or most of the year - Schilthorn).
Hiking trails above 2000 meters are likely to be impassable.
So, for example, you could ride the cable car from Wengen up to Männlichen where there are very nice views, and eat at the mountain restaurant which opens on may 25th
, but probably not hike to Kleine Scheidegg in early June.