The trip to Avalon on Catalina via a ferry is tough for dogs simply because of the requirement that they be muzzled onboard the boat. Other than that, it's no problem.
That said, it's difficult to see much of the Channel Islands outside of Avalon. You really need your own boat to get anywhere but Avalon, and Catalina itself is severely restricted as far as vehicular traffic is concerned. I don't know if any of the jeep tours allow dogs. My guess would be no.
As far as your list of other parks, King's Canyon, Sequoia, Death Valley and Joshua Tree tree all fairly close to each other, but many hundreds of miles away from the Redwoods.
You mentioned Christmas time, though, and there's the big monkey wrench. This year the Sierras (which include King's Canyon and Sequoia) got about eleventyhundred feet of snow. This happens sometimes, and when it does it can be a no-go. Christmas is early for much snow, but it can happen.
Personally, I really like Death Valley but the best stuff would be difficult to access towing a camper that you cared about. Maybe consider detaching it at a campground and driving to Titus Canyon, Dante's View and the Racetrack without it? If you plan on doing any offroading, you'd better have license plates on your bikes. And big tanks.
Joshua Tree, for me, is only appealing if I want to climb rocks. Otherwise, it's no more interesting than any other stretch of uninteresting desert.
King's Canyon/Sequoia is well worth the effort if it's open. Effing big trees, effing old trees and all that.
The Redwoods are amazing, too, but are 500 miles away from the other parks in your list. I guess if you're traveling all the way from the wrong coast it is a relatively minor consideration, but it will take you a full day of travel to get there from any of the other places you mentioned.