I did my first duathlon last month and absolutely loved it! I've signed up for another in October and may do one next month, too. A friend of mine did it on a mountain bike and while she did quite well, she worked much harder than those of us on road bikes. So yes, definitely consider a different bike.

I'm a long-time runner and devoted most of my training to running, although I'm slowly changing that to incorporate more cycling. I'd say to balance both, but others who have more experience may be better able to answer that question. Do try to do some bricks in your training to get used to the bike/run transition.

Good luck.