I do a 22.2 mile commute a couple times a week on the road with some pretty steep hills.
- Get a bike that you can fit a sturdy rear rack to and add panniers for carrying your change of clothes (you will need a change of clothes and shower at that distance). Especially with road bikes, I find that the more deeply bent over position for long rides means that carrying a backpack makes your hands and lower back ache.
- Get a bike that you can easily fit full length fenders on. If you're commuting every day, you're going to be rained on. More importantly, your fancy new bike is going to get gritty/oily crap sprayed all over it's drive train every time it's damp out. Fenders will help this!
- Leave your bike lock on the rack at work (who needs to be lugging around 10 lbs of metal each way that you won't be using??)
- I don't think that padded cycling shorts are a necessity for a 13 mile trip, just make sure that you're comfortable with your seat.
- I carry a spare tube, frame pump, tire levers and multitool with me whenever I commute, but I don't have a cellphone. The frame pump is attached to the down tube, the tools go in a seat bag.
- Invest in tires with puncture proof kevlar lining built in. It's worth it, and you'll flat much less often. Depending on how much debris you've got on your path you might also want to look at slime filled inner tubes. NOT having to stop for a flat is WAY better than fixing one.
- Check your pressure and if necessary, pump your tires before leaving every morning. Deflated tires lead to flats.
13 miles in under 45 minutes should be doable depending on temperature, traffic, wind, and hilliness. I've been doing my 11.1 miles in the hills, snow, and blowing winds during the winter at between 45 and 55 minutes (one day it was 60 minutes, but that was with a 30 mph headwind most of the way and ski pants on).
As far as bike recommendations go, I'd look for something like a used touring bike or a cyclocross bike . . . (Surly LHT or Cross Check both spring to mind). They tend to come with clearance for bigger tires (easier on your hands if your route is full of potholes), clearance for fenders, mounting points for racks, etc. Absolutely go in to a bunch of bike stores and try things out to make sure you don't end up with somethign too small or too big.
My other recommendation is that you ease into this exercise . . . try commuting one day a week, then two then next, then three the next, etc. 26 miles is a long haul when you're not used to it, your body (ass especially) will need to adapt. Drop handlebars are great because you can keep changing your hand position all the time to avoid fatigue . . . but that habit takes a little while to develop if you've only used flat bars your whole life.
Personally, I go way faster on my road bike than I used to on my mountain bike with slicks, or than I do on my hybrid. Being more stretched out and having the drops to duck under the wind makes a big difference. I think you're on the right path by looking at road bikes.