1) What days, exactly? (there's a reason I ask)
2) >"He took the train to the north to the coast and took a charter boat excursion." SOUTH to the coastal town of Seward, the tour boats headed out of Resurrection Bay to Kenai Fjords National Park. Whales, porpoises, puffins, glaciers, mountain goats, eagles but check and call for who is still operating this late, if any one (there are a dozen ships a day in summer). Seward is 2 hours from Anchorage – a spectacular route along Turnagin Arm and through the mountains. There is a bus (van) to Homer http://stagelineinhomer.com and the train does go to Seward, but infrequently in winter. Portage Glacier is on the way as is the wildlife rescue center in Portage.
3) there are hostels in Anchorage, Seward, Homer and numerous other places. $20-30/ night versus $50 for seedy hotels and $75+ for decent hotels.
In Anchorage itself the Museum of Art of History is good, the Native Cultural Center (NE of town) is quite nice, the UAA has various exhibits but not the art / cultural museum that UAF (Fairbanks) does. I recommended the zoo (open all year) for the end of your trip. Whatever critters you didn't see in the wild (or crossing the street in front of you) are at the zoo – from mink and wolverine through moose and musk ox. Polar, black, brown bears, caribou, etc.
Beyond that, it depends on what gear you bring or can borrow – cross-country skiing, snowshoeing, ice fishing, etc. Driving north, especially after sunset (about 4:30 pm) increases your chances of seeing the Northern Lights. Better yet, watch the forecasts a few days in advance:
http://www.gi.alaska.edu/AuroraForecast
http://spaceweather.com
and go for a drive when the forecast looks good.