Hi Wanderdiva -
You have gotten excellent advice from earlier posters.
It seems that you plan to take the walk from Männlichen towards Kleine Scheidegg. It is easy enough to be nicknamed "The Grannies' Walk" while at the same time being one of the most spectacular trails in the Alps.
I've found that going in that direction is a substantially better experience than hiking in the opposite direction. In fact, the trail is not quite level, which makes it a bit easier to walk toward KS. However, the altitude change is minor - from 2230 meters elevation at Männlichen to 2061 at Kleine Scheidegg over about 4 1/2 km.
The walk toward KS has you facing the famous North Face of the Eiger. It grows over you as you walk toward it. The visual and psychological effect is powerful. If you do it in the other direction, you'll find yourself constantly turning around to look back for the good photos.
Many people walk that trail in running shoes, or, maybe, training shoes. Officially, it is designated as the kind of trail that does not requir hiking gear. (More on the below).
The terms are not precise. . Most of the other trails mentioned can also be done in good running shoes...not the racing shoes with thin soles, but shoes with substantial, thick and grippy soles.
However, the advice here on gear and clothing is sound and there are good reasons for it:
And, there is a reason for it. A couple of attached pictures make one point.
The trail starts out almost level. It has a gravel surface. Along the way, there are short stretches with moderate descents for a few meters. Those surfaces demand care, because your feet can slide out from under you on the gravel. Without ankle support, you can easily turn your ankle, and if you are on a trail with uneven and rocky surfaces, it is very easy to do that. At a minimum, heavy, sturdy hiking shoes with deeply patterned, maybe lugged, surfaces on the soles help with the grip, but not the ankle support.
And, when you go down even those modest stretches, a hiking staff lets you brace against it as you walk downwards. Keeps you from slipping. A hiking staff (or two) also helps you with balance on irregular surfaces and even easy climbs.
Also, especially in the Spring and not so much in September, there are places where the trail can be quite wet. If your shoes are not waterproof, you'll regret it.
You can rent such equipment. There a sporting goods stores all around the area which provide that service. I've gotten to the point where I always wear my boots while walking even on paved roads. I'm much more comfortable in them.
The walks mentioned near Schynige Platte might be rough enough that boots or sturdy hiking shoes with lugged or grippy soles would be very desirable.
One way to make the judgement about trails is to use the trail rating system of colored markers and maps.
This explains it:
The red marked routes are so-called "Mountain Routes" and proper gear is recommended; I consider it necessary.
This map is very detailed. You won't need that kind of detail for your walks. However, if you like maps, you can have a lot of fun with this one. And, it shows altitudes.
I have zoomed it in the Jungfrau region, and turned on the option to show hiking trails. They are color coded. Voila!
map.geo.admin.ch/?topic=ech&lang=en&bgLa yer=ch.swisstopo.pixel karte-farbe&layers=ch.bav.ha ltestellen-oev,ch.swisstopo.swiss tlm3d-wanderwege&layers_visi bility=false,true&E=26 35718.34&N=1163309.01& zoom=5
The hikes you have mentioned are "yellow." Officially, suitable for ordinary walking shoes.
The trails around Schynige Platte are "red." Boots and hiking staffs are necessary.
Finally, the Swiss are avid walkers and hikers. Don't be surprised if you find them going uphill past you without notable effort, while you are walking slowly because you need to catch your breath at the high altitudes. And, the trail sign posts are marked in time to the next destination. The Swiss don't seem to notice whether the trails have major altitude changes...but, I do. I match the times on level ground, unless I'm taking a lot of pictures, but I can double the time if the trail has a lot of altitude change.
You've picked the best time of year, with, on average, the best sightseeing weather.
You mentioned Bern in your contingencies.
It is a good one:
A few photos attached.