1) How should I approach breakfast? - eat a full breakfast at 4:15 am, have coffee then go to the bathroom. Eat a small box of raisins as you shut the door.
2) Is there a good tire to use for commuting? I just started using Specialized Armadillos. they are hard t put on but great to ride on. Slick, high pressure, wide enough, and surprisingly comfortable. Be prepared to change a tube or patch a tube hole, including a small pump. Small pump won't get you up to high pressure - but will get you ride-able pressure.
3) I am not sure if I get a locker that I can store stuff in, so what would be a nice bag/backpack to wear while riding? That two different things to consider. Storage at work for food and clothes. Find a nice plastic bin with a cover and ask your boss if its OK, and ask where you can leave it. It should be somewhere convenient to your showering and changing. Nothing precious - mischievous hands are everywhere.
For the bike - get a large rack, with a matching large or expandable rack trunk and a rain cover.
Also, learn to wear less clothing.