I've just got to add to this to maybe save the OP some trouble.
First, I'm assuming you're in New Brunswick, CA, right? You're North of Maine, US? OK, I think you get some significant snow up there and it isn't all of the nice powdery stuff that flies around in the mountains.
After the last winter, I now use the term White Devil in place of the more generic "snow". I have a fabric covered storage building that is shaped like a Quonset style, ie. it has steel hoops that are almost an inverted "U". I was remiss on removing one heavy lot of the White Devil and when it again came down, so did the fabric. Your structure should perform about the same or even worse. Worse, because you have those runners on top of the rafters that are perpendicular to the slope. That will make a perfect compartment for the snow to fill and hold on top of your tarp. If it was me, I'd remove the plastic tarp and add some thin sheeting, fastened to those runners. I'm thinking 5 sheets of 1/4" hardboard and a quart of varnish or paint is around $40-$50. Put the plastic back on the hardboard. This will get you by for a few years, is cheap, and you won't have hardly the trouble as the tarp over the frame.
I wouldn't chance leaving the roof supported by the lagged fasteners. They may hold but the risk would have me doing it different. It sounds like you are trying to accomplish all of this by making the structure fall under the "temporary" building code classification as opposed to an addition to your house or a permanent out building. Since you're familiar with the post arrangement of post on block with steel bracket, I would suggest to duplicate that on the house side. That would make your structure free-standing and may still keep it "temporary" in nature. For extra stability, you could anchor the post base blocks, using an arrangement that will keep to the temporary aspect (able to remove the base block). If you can add a few pieces of lumber to tie the corners to a post or all of the posts together, you'll increase the stability of the structure by 40%.
It's awful easy to analyze and spend another blokes money from a keyboard, just offering it up here so you don't have a big headache. Good luck with your addition!