A person with an ice ax and crampons, the knowledge to use them, and a general idea of the terrain and where the summer trail goes, I would image would be fine. I would not recommend doing it alone, just from a safety perspective. The dog is out of the question. You’ll be crossing or climbing many steep snow fields that are old hard snow where a dog’s paws could find no purchase. You’d have to turn back before jackstraw.
All of the north face trails are going to have snow and ice to contend with for some time still, at the moment, starting just above 8k ft. Your best bet for getting up high is definitely the momyer Creek trail taking it straight up instead of turning toward Alger Creek. There’s a steep face on the east side of Alger creek that is probably still holding a lot of snow. Vivian will be snowy still by halfway camp, I’d assume.
If you’re looking for a good snow free hike and a night on the mountain, you could go from forsee trailhead over to John Meadow then up to join the San Bernardino peak trail, dropping down to columbine camp. It’s only maybe a 12 mile round trip, but it would be doable with the dog. Not sure about snow at columbine, haven’t been there in a while, but it’s a beautiful spot after winters like this past one, with seasonal streams that most people never get to see.