Good hike - challenging but straightforward.
At a slow pace with breaks, it took about 2.5 hrs to get from the Dog Mtn Trailhead to the Dog Mtn Summit using the "less difficult" route (turn right at the Y), then 2 hrs to get back down.
Trail is well-maintained, with an even forest floor most of the way, no steps or big roots. Steady incline almost the whole way up. There are signs, easy to follow. There is not a high number of viewpoints on this trail, but a viewpoint about 1/3 of the way up, plus the last bit toward the summit should show some nice views (it was too socked in for me to see anything).
There was snow starting at ~2000', then ~7" at ~2300', and at least a foot near the top. I used microspikes and poles which was perfect. You could do it without, but I would advise shoes with decent traction since there are some areas near the top where you don't have a ton of buffer if you slip. Wind was brutal for the last 30 minutes - it was almost hard to walk at some points. With similar weather you would want a good windbreaker, gloves, and ear covering.
From 10am-2:30pm, I passed about 12 parties, mostly twosomes.
Up in the Olympics, Mount Ellinor is one of my favorites. Although it's very different from Dog Mountain, I think if you can do one, you can do the other.