I have done all of these hikes many times. You must be a quite experienced alpine hiker, accustomed to a lot of vertical ascent, to do these and be happy about it. None are "death defying" if you are very experienced and very fit. But strenuous, yes. I will answer your questions here but suggest you pop over to the TripAdvisor forums and send me a private message (screen name KimSanJose) along with your email address. I have a 40+ page guide to the region, written from a Wengen base. We have been there 16 times in 18 years for stays of anywhere from 2 weeks to 3 months.
Re: your timing, it is possible but not likely that many of these hikes will be open. Usually the higher elevation hikes (anything above Wengen or Mürren) don't open until mid-June. There are a number of lower elevation hikes you can do but none nearly as strenuous as the ones you have selected. All that said, I will give you the info just in case they are open.
It is always best to get up on the peaks before noon. Clouds tend to start coming in by early to mid afternoon.
1. Jungfraujoch/Männlichen etc: If the route from Männlichen to KS is open I suggest you do this in reverse. Take the gondola up to M (or walk, but it is a strenuous uphill, an ascent of about 3500 feet over about two miles, very steep) then walk from M to KS. That is the direction of the views. Catch the train up to Jungfraujoch at KS. Come back down and have lunch at KS (food up top not great). Then walk or train down to Wengen. If you get started early you can get up top J late morning. I would NOT recommend walking down from Männlichen to Wengen. It is too steep. You will be very unhappy.
2. Schilthorn. Very unlikely the route up will be open above Mürren. It is also very strenuous to go up from the valley floor. On the off chance that the route is open, I definitely would NOT walk down it. Too steep. I recommend taking the train from Wengen down to Lauterbrunnen then walk up the Valley to Stechelberg. Gorgeous route, will definitely be open. Have a snack at Stechelberg Inn, then hike up to Mürren. This will likely be open and is a gorgeous walk. Not too strenuous, will take about 2.5 hours. The views from this route are much better than the views going up to Mürren from Lauterbrunnen. Take gondola from Mürren to Schilthorn. Admire the views for one revolution (one hour) along with a drink. Food here is not good, so go back down to Mürren and have lunch on the balcony at Hotel Alpenruh (right by the Schilthorn gondola station). You could then walk (or train) from Mürren to Grütschalp, then cable car down to Lauterbrunnen.
3. SP to First. This may be open, but again it is rather unlikely. If it is you should expect to encounter some snow at the higher elevations. This is a strenuous hike but not as steep as the routes up to Männlichen or Schilthorn. Time this to arrive at First no later than 4:00, otherwise you will be faced with 5,000 feet of vertical descent. Not a pleasant concept after a full day of hiking. It will take you a couple of hours to get to SP from Wengen so factor this in.
4. Eigergletscher. I prefer to do this in reverse, otherwise it's a lot of descent (which just beats up your knees and gives one little exercise) and the mountain views are to your back. Take train from Wengen to Alpiglen and hike up to Eigergletscher. Then train or walk down to Kleine Scheidegg and train back to Wengen.
5. Boat: an outing on Lake Brienz is very lovely and even good on a rainy day. Good option if most of the trails are closed or you are tired one day. Which hike to skip? Depends on what is open.
Another idea for you is Meiringen to Grosse Scheidegg. Stunning hike. The top bit may not be open yet but there is bus service so you could go as far as you can then hop the bus back down. Lots of detail on this (and many other) hikes in the document I mentioned at top of this post.
6. Pass: absolutely. Buy Half Fare Cards at Geneva station (or wherever you enter the country) for 120 CHF per person then you get half off of all transport. The transports you will need to take to do what you want to do are the most expensive in all of Switzerland. You will save a bundle.