We were in Murren 2 years ago just a few days before Christmas. We’re not skiers, but we had a blast on the sleds! Rental facilities are set up near the Schilthornbahn for the sleds and any other equipment/clothing you may need.
The was an early large snow and conditions were perfect. 3 nights is actually 2 full days which should probably suit you just fine.
Will things be open? Well, your hotel will definitely be open :-) don’t know about anything else, but I would really doubt it. Try to choose a hotel with a restaurant so at least you’ll have a place for dinner without having to wonder if the restaurants will be open. If you arrive from Lauterbrunnen and Grutschalp and there is snow, be aware that there are free luggage sleds (large) to get your luggage to your hotel. Just ask for the key at the station ticket counter and unlock the chain and take a sled for your luggage. Then return the sled of course.
We’re in Murren every year, but that was our first winter time there.