Even though possessing and consuming Marijuana in Alaska (above the age of 21) is legal; to carry it while traveling (person or mail) is illegal. It is illegal to transport marijuana across international borders and so you can never travel with this herb. Note, that if you’re traveling on one of Alaska’s famed small bush planes, you won’t encounter security screening, so the question doesn’t really apply there.
No, you cannot fly anywhere with marijuana. Commercial airports and airlines are under Federal jurisdiction. Marijuana remains illegal under Federal law and the TSA does not respect local law. Where you are flying does not matter as the marijuana becomes illegal as soon as you enter a commercial airport. The TSA may just warn you, but they also could prosecute you.
Yes, as long as you don’t leave the state at any point you should have no problems having marijuana on you while flying. If your flying in state the majority of the time you may be on a small bush plane with no security screening either way. TSA does not bother with marijuana related substances unless you have broken a law. For example if you were in a state where it is illegal.