Hrrm, second post and it is one I am a bit of an expert on.
I work in hospitality up here in Anchorage, (and formerly Fairbanks) so I've seen a lot of hunters through. As for getting your guns up to Alaska, get a hard-sided gun case (The $34 walmart special works) and put TSA marked locks. (TSA locks are able to be opened by a TSA masterkey, which means they won't have to saw off your locks or hold your firearm if there is a problem.)
Remove the bolt from your firearms and merely place them aside in the padding. (Not required, but gun savvy TSA peeps will respect it, and unsavvy TSA peeps will feel good about it.) Take a picture of the case and contents before sealing it. Stick a copy of the picture in the case.
At the airport, skip the kiosks, and walk up to an actual person. Inform them that you are Flying to Alaska on a hunting trip, and have guns you need to ship up. After setting up your ticket, they'll inform you to head over to the TSA baggage area, where you'll do your inspection and handover of the firearms. This should be harmless and quick assuming you've followed the above. Fly as normal.
Generally firearms will be the last thing put onto a plane, and usually are also the last thing brought off. They don't put firearms on the belts, instead they bring them to the TSA baggage office where you present your claimcheck.
In Anchorage, the Hilton Downtown (Not the Garden Inn, Hampton, Homewood Suites, or Embassy Suites) requires you to check your firearm into their gunvault. They have a 24/7 security officer (Or they did when I worked there) who gives you access to your firearms at any hour. (The Marriott Downtown is run by the same company, and likely also has a gun vault.) Most other hotels in town will let you either store the gun in their storage areas, or bring it up to your room. Don't be like the secret service and shoot out a window while up here!
In town, I like Wild West Guns and The Ammo Can for 10mm ammo. (I own a Witness in 10mm) For other ammo there are lots of places, though I got my aught six hunting ammo from Gun Runners.
Any other questions I can answer for you?
What Not To Do: Do >NOT< spend too much money on your case, ESPECIALLY if you will be flying in. The charter flights have very tight weight limits, and a 20lb guncase is not only overkill, but simply too heavy. I really do recommend the cheap black plastic cases that you can bang around and don't care if they get destroyed. They'll protect your firearm just fine.