Congrats on getting started with your commute

1. Sore butt - it takes some time to get used to the saddle. Padded bike shorts can help.
Carrying stuff - definitely get something to carry it on like a rack and panniers that's not on you. MUCH more comfortable.

2. Headlight - definitely a good idea. How bright you need depends on whether your commute is mostly urban or has some really dark roads too.
Wet bike - I wouldn't do anything to it. Chain lube will come off more easily in the rain so you want to check and relube as necessary.

3. Time - Yes it will get shorter as you get in better shape particularly as you are able to ride up the hills you are walking. Even riding just a little bit faster than you would be walking helps.

4. Gearing - If you are unable to comfortable spin up the hills you have now in your easiest gear then yes you can benefit by adding easier gears. Looks like your crankset is 48/38/28. You could also change the 28T chainring to a 24 to get easier gears, rather than change the cassette.