Hey E,
I was over there, the last time was ....let's see, five years ago. I went over 7 winters guiding and I've always stayed at the Matterhorn Hostel http://www.matterhornhostel.com/en/index/ Spartan and cheap...but I was only needing a bed when not on the Matterhorn, not entertaining family. It's 10 minutes up the hill from the train station past the Fallen Mountaineers Memorial and 5 minutes from the lift.
It's Zermatt! They rent Everything! You gotta dig alittle to get to the right shop but they did have carving rentals there in a couple of shops...........Kessler, Virus and SG boards is what I saw.......but the deposit was Huge, like new board Huge! and they charged your card first and then reversed the charge when you brought it back.......could all be different now.
Jan. does sound like shoulder season. I was always there in Feb/March I see you planning a year ahead so you got plenty of time to firm up intel. Good hunting!