1) I'd like to make it a week long trip. I was thinking of starting in Crescent City and ending in San Fran (412 mi). I average without gear around 18.5mph and ride 100 miles a week. Based on that is this trip doable in 5-6 days time?
As John says, six days is doable, but seven would be better. If I were to do it in five, I'd start at Arcata/Eureka. You'd miss Redwood National Park, but be less stressed for time. (And to note: Redwood NP isn't the only place you'll see big redwoods en route!)
And as will be noted, you definitely will go a bit slower with gear.
2) Best way to get gear from Atlanta, GA over to starting point. I'd like to bring my own bike. Is it better to ship it or fly it out?
Are you planning on flying into Portland (PDX?) If so, how do you plan on getting to Crescent City?
You can ship a bike via Amtrak Package Express for maybe $75-100 one way. You can have it shipped from Atlanta to Union Station in Portland, then back from either Oakland/Jack London or Emeryville in the Bay Area.
3) Camping on the coast, good idea? It sounds nice, but am I going to be fighting crowds of people to get a good spot? Better to reserve a camp site or just wing it?
Great idea! The crowds don't come out until after Memorial Day, and many of the places you would camp on the route have specific hiker/biker sites.
4) In regards to the above ^, bivy sack and a ENO hammock enough to get by? Probably a rain tarp too?
You probably won't go hurting for trees, but it's wise to have a non-hammock backup, esp. if you run into any hammock prohibitions. If your hammock doesn't have a rain fly, get one.
5) Outside of regular riding clothes and tools, any essential gear that you think is crucial? I was going to bring a JetBoil with for coffee and food, rain gear, and a book.