Here's a picture of Sara and Brent N this year on the JMT between Yosemite Valley and Half Dome.
Hi again Sara! I hope we can answer most of your questions here, at least the ones for the California section of your trip.
As for hiking from Sequoia to Yosemite, the JMT is pretty much the one and only way. If you are thinking about another trail, there isn't one. That is partly due to the terrain. You see, all the drainage is east to west toward San Francisco, so a trail going between the two national parks would have to climb every ridge and drop into every canyon along the way. The John Muir Trail and the Pacific Crest Trail (one and the same for most of the JMT) travel as close as possible to the crest of the Sierra Nevada range, so some of that up and down is eliminated.
If the southern extent of your trip is the southern end of the JMT, you will be happy to know there is public transportation from Lone Pine (just down the road from the JMT terminus trail head) back to Yosemite Valley, or north to Reno, Nevada.
Public transportation to Sequoia N.P. is available in Visalia, California. (Sequoia Shuttle, I think.)
As for the Crater Lake area, I am not sure about public transportation. But I might be able to connect you with someone who lives in Eugene, Oregon, (about 3 hours from C.L.) and who also bicycled that section of Highway 1 down the west coast last summer.
