I must strongly recommend a mountain bike. Your comfort and easier gearing are worth more than the extra speed of a road bike. You may wish to purchase a road bike in the future, but for now, a mountain bike will be more enjoyable.
I have found:
1) Road tires do not work well on dirt/rough roads. Poor traction, affected by ruts and they will get ripped up quickly. You can't work around this. Mountain bike tires work wonderfully on bad surfaces, well cushioned and easy to steer.
2) Seating position might be uncomfortable on a road bike? It's not for everyone. You must decide how fit you are for this. There will be much weight on your wrists, you probably haven't experienced this yet. Your neck might be a problem too, as you will be looking up for long stretches (scanning the road ahead).
3) Road bikes are much faster, knobby mountain bike tires are, by their nature, dreadful on highways. You can chose treads that will work pretty well on paved roads.
4) You should definitely do this. Cycling is my favorite sport, at my tender age of 47, it's the only sport left where I can sweat alot and not hurt myself.