First time hiking the AT. Planning for July 28 - Aug 3. NOBO

Driving from Michigan and staying at Franconia Notch State Park Saturday night. Waking up at 5am Sunday and getting to the trailhead by 6am.
First question: Parking at Liberty Springs/Whitehouse trailhead. Can I leave my car here for the week? Will I get a parking space Sunday 6am?

Hiking to Garfield Shelter (10.1 miles) first day.
Second question: Does this shelter have tent camping? How is it?

Garfield Shelter to Zealand Falls Hut (9.1 miles) second day.

Zealand Falls to Ethan Pond (4.8 miles) third day.

Ethan Pond to Naumann Tentsite (8.3 miles) fourth day.

Naumann Tentsite to Madison Spring Hut (11.5 miles) fifth day.
Third question: Too far? Didn't find a good stop before the hut.

Madison Spring Hut to Joe Dodge Lodge (7.1 miles) sixth day.
Then take shuttle to Highland Center for the night...shuttle to car next day.

Hiking background.
2015 - 100 miles (10 days) at Philmont Scout camp. Including day hike to top of Mt. Baldy (12,400 ft)
2017 - 60 miles at Philmont. Mt. Baldy again.
2018 - Cumberland Gap 4 day ridge hike. (cake walk)

Thanks for your help.