First, I think you or your autocorrect meant to say "Grindelwald" and not Grunewald. Spelling is important when you get tickets or you may end up in Gimmelwald, miles away from Grindelwald.
If you only have 4 days, I wouldn't go up BOTH to the Jungfraujoch and Schilthorn. I'd pick the latter, espec. since you want to hike down, plus there's too many good trails to explore instead of spending half your time on mountain tops. Speaking of hiking down, know your limitations. Hiking down sounds so easy, until an hour in and your shins start aching like crazy. It's not just an age thing. I walked down with my 14 yr. old from Kleine Scheidegg to Wengen. It looked like a nice 1.5 hr downhill walk, but we both had aching shins. So my advice is to hike down not from the Schilthorn, but from Birg to save your legs a bit.
As for biking, there are certainly bike-authorized trails in the area, and the Tourist Info office in Grindelwald should have a map of such. I've seen it before. I know Rick Steves shows renting a bike (I think in Murren) and it is on one of his free videos on this site. However, here's something to consider: the scenery here is the BEST in the world, so I'd rather spend the majority of my time looking up and around and taking pictures. When I ride a bike on a trail I spend the majority of my time looking DOWN for stones or ruts or whatever to make sure I don't break my neck. So I prefer walking (at least HERE) to riding. You mention biking between Grindelwald and Lauterbrunnen, and I suspect you can, because there is a road and trail between the two. But I would never take it, because you will be out of sight of the mountains--you'll just see trees and a stream. If you were there 2 weeks, ok, but given just 4 days, I wouldn't spend any time away from the fantastic panoramas, unless they were fogged in.
You mentioned hiking, and here are the best hikes, in order from best to still very good.
1. Mannlichen to Kleine Scheidegg, about 1.5 hours. Save for the best day.
2. Schynige Platte to Bachalpsee to First to Grindelwald (last leg via cable car). This takes 6 hours so leave early since cable cars stop at 6pm. If you don't have that time, or weather is iffy, then go Grindelwald to First (cable car) and take the 45 min walk to Bachalpsee (Google image it).
3. Lauterbrunnen to Grutschalp (via cable car) then walk to Murren. From Murren you could go up to the Schilthorn, but check first if the mountain top is fogged in. There's a tv monitor in the Murren cable car station that shows the top. If fogged in on top, you could walk up the trail going up the hill diagonal to the cable car line, and end up at the restaurant and cows at Spielbodenalp.
4. On a dreary day you can walk or take the Postbus from Lauterbrunnen to Trummelbach Falls (Google image it). You can even see those falls in the rain, since they are inside the mountain.
All of this makes more sense when you study the maps:
http://ontheworldmap.com/switzerland/ski/jungfrau/jungfrau-summer-map.jpg
http://www.gimmelwald.com/pics/gimmelwald/hikemap.jpg
Good luck!