Yup, as MichaelJ says - I just did the loop up the easy way and down the steeps, that's up on 19 Mile Brook, Carter Dome to Zeta Pass, along the ridge to the Dome then down the steeps to the hut and out on 19 Mile Brook.
As far as "How long?" It took me a shade under 9 1/2 hours including all stops, picture taking and solo trailbreaking with too short a snow shoe (I *should* have taken the 30's <sigh>) I'm slow so this is a worst case scenario time
Everything depends on the weather, but even if reports say the trail is broken out I'd take snowshoes and crampons. You *could* bareboot the steep section up/down from the Notch but crampons add a good measure of safety.
I bypassed Height on the way up because it was completely unbroken, figuring I'd do it on the way back. Then I changed plans and went down to the Notch and out on the 19 mile Brook trail.
19 Mile Brook gets a lot of traffic because of the hut so is almost certainly going to be well packed out. I bare-booted 19 Mile Brook all the way in and out, barely left a mark
A great loop (well, lolipop really) hike I'll probably do over many times.
If you don't mind seeing previews of the trip, my pics for this hike can be found starting
HERE
Bob