Hey Bob, I'll try to help you out a little bit. Assuming you are in shape, this should all be possible. I will emphasize though that being in shape isn't everything - the terrain in the Whites can be extremely rocky and tolling...the best practice for hiking is hiking. Day 1 definitely seems doable. For day 2, I haven't hiked this trail, but your plan for seems a bit ambitious (IMO). From what I've read, the area around Garfield/the Twin Mountains is very rocky tough hiking. It looks like this will require you to summit South Twin, Guyot and Zealand which adds up to a lot of elevation gain. A better plan might be to stay at Zealand Falls Hut or the Guyot Shelter. Day 4 looks alright but keep in mind you will be going up and down a lot. The weather can also be pretty fickle in the Presidential Range so I would have an "escape-route" planned out. Avoiding tourists on Washington will be impossible in July, but there is definitely no shortage of alpine terrain to explore.
Don't know anything about the shuttle.
It might help if you give us an idea of what kind of hiking you have done and what you're comfortable doing. A strong hiker could do what you have suggested, but without knowing your capabilities it's hard to give you the best advice.
If you don't already have them, I would pick up the AMC maps of the White Mountains (sounds like these two should cover your trip:
Franconia-Pemigewasset,
Presidential Range
The
AMC White Mountain guide is considered the hiker's bible for the whites...highly detailed trail descriptions/everything you need to know.
As a side note, I just ordered this book and I haven't read it yet, so I can't recommend it with any certainty, but
The Country Northward sounds like it tells the tale of a backpacking trip very similar to yours.
Hope this helps. Happy to answer any other questions you might have...sounds like you're going to have a really great trip.