The snowiest parts of this route are Silver Pass, Seldon Pass, and Piute Pass, along with all other high elevation areas. You best source of information about current conditions is going to be hikers who you meet on the trail, along with web sites like trailjournals. The forest service can also provide a general snow level (for example, snow about 9000 feet elevation), but this is just a general guideline. In early summer, you won't be able to find the trail and will need to rely on your topographical map. You also may need crampons and ice axes (later in the season, the snow will be soft enough that you won't, but again, ask other hikers what the current conditions are like).
Generally speaking, if you can deal with the snow, your dog can too. Having four paws is an advantage and bare paws have pretty decent traction (see the gear page for more discussion of booties).
October snowstorms can be deadly, if you get caught miles from a trailhead, no longer able to see the trail, and with knee, waist, or higher, powder to wade through. Consider planning your hike earlier in the year (or taking snowshoes, although they are heavy and bulky and might not suffice for a large accumulation of fresh snow).
Wading through knee-deep water is common, especially during snow melt season (which can last into June or July). Be careful - stream crossings can be dangerous. If you aren't comfortable with a crossing, go up or downstream until you find a log fallen across the stream, or backtrack to a different trail. Also, ask hikers coming the other way about upcoming stream crossings - they may be able to tell you about a log, a shallower place to cross, or some other tip.
Here is what has worked for us in crossing streams with our dog:
1. Assist Handle. One way to cross a stream is to have your dog beside you as you hold the assist handle of their dog pack. This technique works well as long as you have a solid footing and your dog doesn't mind being carried (perhaps through a place where their feet cannot touch the bottom).
2. Extendible leash attached to pack (not collar). This allows you to give some help to a dog which might be starting to get swept away by the current. The advantage of the extendible leash (as opposed to a short fixed one) is that the dog need not be right next to you (for example, you are in shallower water or on the bank).
3. Let your dog figure it out. Dogs are better than we are at climbing across the tops of submerged boulders and jumping from safe spot to safe spot. Because they have four legs, they are less likely to lose their balance. So if your dog seems to have the desire to find their own way across, often it is best to let them.
Your dog will improve with practice. There are also agility classes for dogs (although I can't really comment on their relevance for backpacking, as our dog has learned his skills on the trail).
Your dog will need to be able to deal with horses on the trail, as they are commonly found on most parts of the Canine John Muir. Teach your dog to be calm around horses (and humans and other dogs, for that matter). Practice on day hikes if you can. Getting your dog used to horses, and restraining your dog with a leash, can prevent a chain reaction in which your dog is over-exuberant, a horse gets spooked, and someone gets hurt.