You are asking a lot from one bike - commuting on rough roads, triathlon and cyclocross (CX)!
I would choose a hybrid bike (like a Cannondale Bad Boy, though there are many to choose from) based on the following characteristics:
- Larger 700c wheels that make them more efficient/faster than a Mountain bike.
- Wider rims and tyre choices that make them stronger than Road/Racers.
- Flat handlebars similar to a Mountain bike for a more upright position.
- Good range of gears to handle varied terrain.
- Often have suspension forks for comfort (though if too cheap, can be more hindrance than help).
- When fitted with disc brakes, have good stopping power.
A hybrid bike would be a good fit if the majority of your riding is going to be commuting and comfort is a key driver in your decision. It will be quicker than a mountain bike (assuming 26" wheels and knobbly tyres), though not as fast as a road bike.
A road bike would be faster still, but a more aggressive aero riding position to get used to and you'd probably wreck the wheels riding CX. A CX bike could give you a position similar to the road bike, but is also designed more with racing than comfort in mind. I think CX is a bit more specialised and therefore I speculate you are likely to find less choice and possibly higher prices compared to other bikes of a comparable specification.
You can choose the best tyre width/tread/compound and pressure for the job at hand:
For roads and cycle paths, a smoother rolling tyre inflated to a higher pressure. Possibly some puncture protection if the roads are on the rough side.
For triathlon, you could select narrower slick tyres pumped even harder for less rolling resistance. Perhaps consider some clip-on aero bars. I've seen a right mix of equipment out there - people on very cheap full-suspension bikes with knobbly tyres doing triathlons, but I think they like the challenge and make things hard for themselves!
For CX, you could pick a slightly wider tyre with a more knobbly tread and perhaps run at a slightly lower PSI if you need more grip.
Hope this helps!