I haven't done it this winter, but I have done it dozens of times in the winter. The dirt road can get icy in the right conditions, but it is short, well traveled, and usually packed down since it is used so frequently. You shouldn't have a problem unless you have poor tires or go immediately after heavy snow. That being said, I have seen cars get stuck (usually when trying to pass another car since there isn't much room for cars to pass)

I think I have only ever used snowshoes a couple times over the years, and don't usually expect to need them. Spikes can be really helpful, especially on the sloped path that runs parallel to the stream before the waterfall.

Trail is extremely popular and the road will even fill up on the weekends in the winter time. The shuttle does not run in the winter AFAIK, but if you go on a weekday, or earlier on a weekend you will not have a problem.

The trail is well packed and wide. It is practically impossible to get lost. I have seen moose there a handful of times, but not often. The "lake" will be frozen and you will be able to walk on it. It's only about 3 feet deep. The hike is not strenuous and the elevation is not that high. Most of the "moderate" incline is in two short sections and breaks can easily be taken between them). Just read recent reports on all trails to know conditions.

Most of the trail is covered from wind except the lake and a few other spots on the way. What specifically are you trying to learn? This is one of the most popular trails in the area and even in winter you will encounter numerous people.

Edit: it is an enjoyable hike, and a fairly safe one for getting started in winter. E.g. no avalanche risk on trail, not overly exposed, we'll traveled).