Don't get a fixie. That's just hipster talk. They are simple and efficient but 5 miles each way will be hard - especially with a few hills. You want gears and a freewheel.
Used gets you a lot more. Craigslist is a good place to look.
It it rains where you are you want decent mudguards - proper bolted on ones make a huge difference.
That and good quality slick tyres - some puncture resistance is good but not the really heavy ones which feel sluggish. I like the Schwalbe Marathon range - not had a puncture on my winter bike in 3 years of using Marathon Racer's. Change them when they start to look worn or you'll start getting flats (like me yesterday - new ones fitted today, I knew I should have done it last weekend...).
Learn how to change a flat - 5 minutes when you know how. Get a little kit and a decent mini pump.
Type of bike is up to you - either a MTB with slick tyres or a road bike. A road bike will be faster but the position is more aggressive and they're a little less durable.
I often recommend a spare bike. It's really handy to have something else to ride so if you have to fix something - or even go to ride and find a flat tyre - you don't have to rush it. A cheap spare is really useful.
Maintenance is simple. Make sure your drivetrain is clean and lightly lubricated, change parts when they wear out and the bike will last years.
Saddles are a personal thing, try a few until you find one you like.
If you have to lock it outside get a big beefy lock and leave it there - you don't need to carry 5lbs of lock back and forward.
