Lola,
It's been awhile since I've lived on the island, so memory is not too clear on the best spots. Sherry had lots of good suggestions.
I do have a few thoughts to add though.....
Given the route you're considering, I'd also recommend going across at the Aldergrove border crossing. It's a smaller station but could also be busy, but it has one advantage in that it provides fairly easy access to Highway 1. I normally use Abbotsford/Sumas as it's the most convenient from my location, and it also provides really easy access to the freeway. Once you're on the freeway, you could travel across the Ironworkers Memorial Bridge to the Upper Levels / North Shore and then to Horseshoe Bay. One point to note however is that on that route you'll be travelling across the Port Mann Bridge and will get nicked with a toll (it's about $5 as I recall).
If you come across at either of the Peace Arch crossings, not only will they be "busier" but you'll face more traffic density from there to Horseshoe Bay. One point that's worth noting is that the GPS maker Tom Tom just labelled Vancouver traffic as the worst in Canada and fourth worst in North America! There's a massive construction project underway on the Burrard Bridge so try to avoid that area and downtown Vancouver.
It's been many years since I've been to the Sunshine Coast, but Sechelt, Roberts Creek and area should provide some nice camping opportunities. You can reserve space in Provincial Campgrounds online, but they typically book-up very quickly. I don't know of any cabin rentals, but Trip Advisor should have some information on that.
You could then come across to Vancouver Island from Powell River and explore some of the suggestions that Sherry mentioned. One area that's very popular is Tofino, although it's a bit out-of-the-way. There's an award winning hotel called the Wikkaninish Inn which has incredible views of the ocean, as well as a gourmet restaurant. Not exactly a "cabin" but I'm sure you'd be very comfortable there! One other interesting point is that they accept pets (must be specified at the time of booking), which includes a "paw washing" service for those romps on the sandy beach, and a pet sitting service. Just a thought......
There should also be some nice outdoor spots around Parksville and perhaps around Chemainus or Duncan (can't remember for sure as it's been awhile since I lived in that area).
After that continue south to Victoria and take the Ferry across to Port Angeles.