Will I find snow in Murren the first week of June? Bike riding and hiking still an option that time of year?
There will not be any snow in Murren the first week in June. No problem if you want to bike or hike.
We were there in early June a couple of years ago. No snow anywhere in Muerren (5,200'). Weather was warm for us but it could easily be raining. The beautiful wildflowers were in bloom all throughout the hills. The farmers were walking their cows through Lauterbrunnen and Muerren up to higher pastures. You could often hear the melodic cows in the early morning as the cows came through. Nothing wrong with early June.
You'll see snow a couple of hundred meters above you, but none at Murren's elevation. Some of the trails at higher altitude may still have some patches covered in snow, but they won't be impassable.
We visited the area in late May several years ago and some trails at much higher elevations than Murren were not open. Fabulous otherwise.