Since I live in the general area and have traveled extensively through these places, I'll add a few observations:
Early December is unlikely to have significant snowfall (though anything can happen)
Walnut Canyon would be on my to-visit list - it's very close to Flagstaff (about 30 minutes east on I-40) and pretty cool IMO.
Monument Valley would definitely be on my list (to at least drive by and pullover, if not actually entering the park), and it should be convenient to your planned travel. If you don't have a preference, I'd not stay in the hotels at Monument Valley but drive north and pick a place in Mexican Hat (or simply continue on to Moab/Arches). There's other awesome stuff near Mexican Hat (like the Moqui Dugway and Valley of the Gods), but you really don't have the time.
Antelope Canyon is amazing but won't be nearly so in December - the "magic" happens between 11am-1pm when the sun is directly overhead, and you don't get as good of an angle in December (see here for a brief description and a few helpful diagrams -
Understanding Astronomy: The Sun and the Seasons). I wouldn't bother.
Please bear in mind that any activities north of Flagstaff may be adversely impacted by the US-89 detour just before Page due to a huge road fracture many months ago - although I believe the detour route is now paved, it's still an extra 20+ miles and not as nice of a road. If you don't need to visit Page, simply take 89 to 160 to Kayenta and 163 (and then 191 to Moab) - or take 89A if headed in the direction of Bryce/Zion.
If you do make it up near Page, I'd definitely stop at the Horseshoe Bend overlook - it's simultaneously amazing and terrifying (to me, anyway). Absolutely nothing to prevent you from falling ~1000 vertical feet to the river below, but at the same time serenely beautiful.
Arches is probably my favorite park - especially at night on a full moon. Canyonlands (the north entrance is more convenient when you have limited time) is neat too.
You can do a great loop that starts/ends in Flagstaff (or at the Grand Canyon) and covers: Monument Valley/Antelope Canyon/Page, Newspaper Rock SP, Arches, Canyonlands, Capitol Reef, Cedar Breaks (likely inaccessible in December), Bryce, and Zion. I never cared for Zion but am an outlier in that regard.
On the way south from Flagstaff, you'll pass by Montezuma Castle, Montezuma Well, and Tuzigoot national monuments - I think they're all interesting but Montezuma Castle is kind of a must-see (and is right off the highway, so very convenient).
In Sedona (if you have time), Red Rock Crossing State Park is beautiful (if overpriced) - a great place for a picnic, a short hike/walk, or some attempts at photography.
In Tucson, if you like science, definitely check out Kitt Peak National Observatory.
I'd second (or third?) the recommendation of Bisbee - it was definitely not a bad way to spend a few hours once I was bored of Tombstone (no offence, but that took all of 10 minutes).
Be aware you'll almost certainly get stopped by a Border Patrol checkpoint or three in that area.
Kartchner Caverns is a state park in the area that gets good reviews - I've never been and am not sure what it's like in December, but it would be on my list if I had time and it was open.
Have fun and be safe!