I am gonna suggest coming for 10 days now and exploring what you can - even if you don't cover everything you'll have a better prospective on places to visit next time.
There are two major paths to explore CA nature - coastal and inland/mountain. Since you plan on spending 4-5 days in SF you can make a trip to Monterey/Carmel and get a sense of the coast at that point.
For your road trip I would focus on Sierra and parks. Crowd will diminish soon as school is about to start.
If you want to stay in the parks consider Kings Canyon or Sequoia as an alternative to often overpriced and overcrowded Yosemite. Cross the mountains on 120 or 4 or 108. You can take 395 north from LA and spent a couple days in Mammoth and a couple days near Tahoe.
I wouldn't put Lassen on my first trip list it's just too far away from everything else and requires lots of driving. Just focus on LA-SF corridor for now.
Consider priceline to get hotels especially in urban areas - I suspect you'll get a nice 4* place for less than what you are planning to spend now.
The point is - if you can come now - do it, don't wait.
Cheers