4 miles is a nice short easy commute. Work out the costs of your current commute. A bike will save you money. The best option is probably a used touring style bike, which has rack and fenders. A decent used hybrid or ridgid MTB will also do the job and the difference in speed wont make a huge difference. A pannier bag (you just need 1) is a better way than a backpack esp in very hot weather.
Over 4 easy miles, the difference in speed between clipless pedals and toe clips is marginal, so you could wear tennis shoes.

You have to think about a short commute differently to a training ride. Cooling down at the end of a hard ride takes time, so ride at an easy pace, and cool down on the bike. You will travel more slowly, but your journey will be as quick.
You dont need to dress like a Tour de France rider over that distance. You can if you want to and have the kit, but its not mandatory.
In cool weather, commuting is easier than in summer heat. You can even ride right through the winter.