Mid-September is just before the fall equinox, so day length is quite close to night length, 12 hours (this is actually true in mid-September all over the earth's surface, not just in Alaska). Day length will be a little bit longer before the equinox the further north you go, e.g. Fairbanks. The Navy's website at
http://aa.usno.navy.mil/data/docs/Dur_OneYear.php can tell you exactly for all of the locations you mention. For the places you are going, and the length of night in September, I don't think it is likely you will see the northern lights. I have seen northern lights in late September, but it was in places like Barrow where night length is significantly longer than day length by the end of the month.
The weather can be OK, such as it was during President Obama's September visit to Exit Glacier outside Seward but snow by mid-September at Denali Park is also not unheard of.
As far as the time you have budgeted, it seems like a lot of ground to cover for the amount of time. If you are driving from Fairbanks back to Anchorage, you can stop at the entrance to Denali Park, but a one day visit to Denali Park from Fairbanks is just a lot of driving and you won't be going very far into the park. Driving Anchorage to Fairbanks takes most of a day (~360 miles). It wouldn't be that interesting with your short visit to do it both ways.