I appreciate all the replies. I didn't have the forum set up to email me if I got replies, and I was all disappointed and thought that nobody had answered me yet, until I came back here and saw two pages of replies.
I analyzed the tracks using some field guides that show you how to identify cat tracks. You can draw various lines across the images and show that the toes are not lined up equally, or that the two outer toes are pointing inward at a particular angle, or that the heel pad is squared off at the top near the toes, and most of my pictures seem to meet the criteria (in my opinion - at least, the pictures that I suspect are cats, and not the other pictures that I'm sure come from other animals).
As for the toenails, from what I've read, I think it's a usually/sometimes kind of thing, rather than an always/never kind of thing. You can sometimes see cat toenails in the right situation. Maybe the snow was deep enough to press against the front of the foot. In a lot of the photos, the cat was going downhill and sliding on the ice a bit, which would push the front of the foot into the snow, so that even if the claws were sheathed, they might leave an impression anyway. Or the poor cat was digging its claws in to stop sliding.
There are indeed lots of dogs that could have walked on that trail. It's a busy, popular trail (which makes it all the scarier to imagine that a big cat was there during the night), and dozens of people walk on it every day when the weather is good, and many bring dogs.
I've actually decided in my own mind that they are a cat, but I will never be 100% certain until I see the cat itself face to face, and I probably would not be very happy about it if that happened, so I will tolerate having a little bit of uncertainty. Enough people in various forums have told me about experiences where they showed things to the game commission and the game commission lied to them, that I believe they're probably lying to me too.
How much does a trail camera cost? Thousands of dollars, or cheaper?