Well, I'd say cbalaska pretty much covered it. A couple things I'd like to add:
1. Yes, that's a VERY aggressive itinerary--by the time you're done, you'll have seen more of Alaska than many Alaskans and certainly more than most tourists. If you like experiencing as much as possible when traveling (like I do), then I think you'll have no problem doing the trip as you suggested (with cbalaska's modifications).
2. DEFINITELY get a copy of The Milepost. It's available at the places cbalaska suggested, but I'd also suggest getting it at one of the Costcos in Anchorage (if you have a Costco card). I think it's something like $12.44 at Costco--half of list price. Sam's Club may have it at a similar price, and WalMart has it for a few bucks more. Even though I've lived here for 11 years and have been on every accessible road in Alaska (except for the Elliott Highway past Livengood, towards Manley Hot Springs), I get a new copy of the Milepost every year. If you want to know what something on the side of the road is, the Milepost will tell you.
3. cbalaska said that traveling the two-lane roads will take you longer than you expect. I'd say that's not as much from poor highway conditions (I've had my cruise control set at 75+ almost all the way from Anchorage to Fairbanks--don't tell the State Troopers--and done the trip in about 6 hours) but from the abundance of scenic views you'll want to pull over at and the curiosities and sights along the road. All of the major roads you'll be traveling over are in excellent condition--which brings me to my next point:
4. If you can cram it in, I'd certainly recommend driving north up the Haul Road, maybe up to the Arctic Circle. The terrain is almost a moonscape in places, and the Arctic Circle is something that not many people get to pass. If you're a reasonably speedy driver, I'd plan on about a three-hour drive each way from Fairbanks, so it'll take much of a day from you, but depending on what your interests are, it may be worth it. HOWEVER, beware that the Dalton Highway north of Livengood (and perhaps part of the Elliott Highway south of Livengood--my Milepost is out in my car) is a gravel road. And as someone who used to work at a rental car agency in Anchorage, I have to warn you that all of the major (chain) rental car agencies in Anchorage consider travel on gravel highways a violation of the rental agreement. None of the rental companies in Anchorage have installed GPS trackers (as far as I know), so it's up to you to decide whether to risk it. If nothing happens, you'll probably be fine. But if something were to happen on the highway, any coverage purchased from the rental agency would be void, and your credit card and/or insurance companies may also decline to cover you as you were not operating by the terms of the contract (you'd need to check with your credit card and insurance companies on that--that's speculation on my part, based on my experience dealing with claims). However, the good news is that in all my years of Alaskan and northern Canadian travel, I've only had one flat tire. It was from a nail--not a piece of gravel--so I don't think you'd have anything to worry about. The biggest dangers are other drivers--the same dangers you encounter on paved roads. If you do drive up the Haul Road, be aware of both oncoming and passing trucks: they hug the middle of the road (in places, the gravel road is humped in the middle for drainage) and have no problem doing 70+ and throwing pesky tourists off their road. Oh, be aware of rock chips in your windshield--that'll be the most likely thing to happen to your car on both paved and gravel roads. (If you take coverage, I suppose you could, er, fib and tell the rental agency that the rock chip happened on the Glenn Highway just outside of Anchorage, but I can't in good conscience recommend that...)
Also, if you do deviate from your plan, be aware that the Denali and Edgerton Highways (east/west from Cantwell to Paxson and east of Chitina, near Copper Center) are also gravel and not covered by rental contracts.
5. The Sheraton does not offer an airport shuttle service. You may want to call them to see if they provide service around downtown, though--I seem to recall seeing a Sheraton van parked at the 6th Avenue entrance. If not, the Sheraton is within easy walking distance (1/4 mile or so) of the heart of downtown, though (it's not as far as cbalaska sort of indicates). It's not in the greatest area (the east end of downtown is a bit run down), but you should be fine if you stay on 5th or 6th for the few blocks until you hit A Street. Oh, and another minor correction to cbalaska's notes: it's more like 7 miles from the airport to downtown and the Sheraton (only because I had to know when potential customers asked me how much a cab fare downtown was). The point is, though, it's within easy driving distance--15-20 minutes tops.
6. While you're walking downtown: If you're in Anchorage on a Saturday or a Sunday, BE SURE to check out the Downtown "Saturday" Market, featuring some of the freshest Alaskan produce (yes, we grow stuff up here--all the way from peas to 100-pound cabbages--too bad you're not here at the State Fair in Palmer), live music, and great "fair" food.
7. I'd also recommend the side trip to Talkeetna. Be aware that it's a spur road, not a loop, so it's 15 miles out of the way each way (that is, a total of 30 extra miles), so plan accordingly. It's a cute little touristy town that probably offers one of the best glimpses of life in a "real" Alaskan "frontier" town. (Of course, there isn't any one thing that can be considered "real" Alaska: for me, "real" Alaska is Anchorage or Fairbanks; for those along the coast, it's Valdez, Kodiak, or Sitka; for those completely off of the road system, it's Nome, Barrow, Shishmaref, Quinhagok, or Iliamna. That's why there's such a distinction between Southeast, Southcentral, Southwest and Western Alaska, the Interior, and the Arctic Slope regions. Each region has a completely different character--keep in mind that they're as big as some states. But I digress...)
8. In many ways, I'd agree with cbalaska's recommendation to take the train to Fairbanks. Full disclosure: I do work for the Alaska Railroad and I've always had an affinity for trains. It's an easy way to enjoy the scenery Alaska offers without the distraction of driving. However, you do lose the freedom to stop and see what interests you, and so I would have to recommend taking the car. (The railroad parallels the highway for most of the way, so you do see much of the same scenery.) If you do decide to go with the train, make sure you check on a one-way rental from Fairbanks to Anchorage. Only some of the agencies do this (Payless, Avis and Hertz come to mind--I don't think National does), and there will probably be a $100-300 drop charge unless the agency is desperate for cars to be returned from Fairbanks to Anchorage.
9. I will say that $435 is a dang good rate for 10 days in Alaska--in peak season, I've seen people paying $80-100 per day for compact cars. As far as whether you need a compact or full-size car--that's going to depend on your preferences. If you travel fairly light and don't mind driving a Chevy Aveo or a Chevy Cobalt (National's compact cars in Anchorage, last time I saw them) for 1,000 miles, then you'll have no problem--again, all of the roads are paved and in good condition, and there's no reason a little car would have any problems. (We took a 1984 Toyota Tercel around Alaska for several years--even all the way up the Haul Road to Deadhorse and out the Edgerton to McCarthy--and didn't have any problems, so any new car is going to be fine anywhere else in Alaska.) However, if you like space and comfort, Hertz's full-size (which will probably be a Ford Taurus but may also be a Ford Five Hundred, if they don't consider that a premium car) would certainly increase the comfort level of your trip. National may also have an upgrade available at a price less than Hertz's, so you can wait and try that route--heck, they may even give you a free upgrade if they're out of compact cars, which happens a lot in the peak season. (Hint: don't look too eager for the upgrade, or they'll see you really want it and may quote you a higher upgrade price. However, National usually quotes fairly low upgrade prices, at least as compared to some of the other agencies, and their commission structure doesn't reward outrageous upgrades more than $2 ones, so play your cards wisely. OK, enough insider information...)
10. cbalaska recommends gold panning in Fox, Alaska, about 10 miles north of Fairbanks. I haven't done that yet, but I hear it's pretty popular. (I just moved to Fairbanks.) The name of the place you're looking for is the El Dorado Gold Mine, although there are other touristy gold mines (and many real commercial mines) scattered throughout the area. Some other things to consider doing in Fairbanks would be visiting Pioneer Park, riding the Riverboat Discovery, and checking out Chena Hot Springs. There are also lots of things to see at UAF in addition to their outstanding museum, but the tours of their research laboratories may not fit your schedule. (See
http://www.uaf.edu/univrel/Tour/ for details.) However, for the time you'll be here, I think you'll be pushing your time in Fairbanks as it is. (Side note: the best place to get close to the pipeline is the viewing platform/information center a couple miles before Fox on the east side of the highway.)
11. I'd agree with cbalaska that the ferry is the best way to travel from Valdez to Whittier. You'll really need to be careful with your schedule, as there is only one sailing per day (and maybe even only one every couple of days). Also, like he said, if you take the ferry, you can cancel the 26 Glacier Cruise. While the cruise will take you past and much closer to many more glaciers, the ferry sails close enough to the Columbia Glacier (one of the biggest and most majestic in the area--as large as Los Angeles) to get your adrenaline going when it cracks and calves, and you're guaranteed to see awesome sea life on the way.
12. I'd also agree with cbalaska on your Days 7 and 8 plan. No sense going all the way to Kenai when your activities are going to be in Seward. There's really nothing to see in Kenai, so you might as well stay in Seward that night and then go to the Kenai Princess (in Cooper Landing--don't get confused) the next day. (The Kenai Princess is pretty much at the edge of the scenic part of the Sterling Highway--any further and you get into rolling hills and flatlands of forest. You may want to go a few miles down to the confluence of the Kenai and Russian Rivers and see real Alaska "combat fishing" (shoulder-to-shoulder) in action--that's always an interesting thing for Outsiders to see). Besides, with the activities you want to do in Seward, you'll need both the night of Day 7 and the bulk of Day 8 to do them.
Also, take a peek at my other posts in this Alaska forum, too, here:
http://www.flyertalk.com/forum/showthread.php?t=559223 -- it covers some similar territory.
All right, I think that pretty much covers my opinions. Have an awesome trip! You'll remember it for a lifetime.