1) Proper Cyclo-Cross bikes are designed to race on. They are very rigid and very harsh. This means they accelerate very quickly, but are uncomfortable on long rides (not just sore arse, also numb hands from vibrations etc). Road bikes are differentiated by out and out racers at one end, to slack, slow tourers at the other end. For commuting purposes you want the moon on a stick, ie something that is fast and responsive, is also comfortable and is flexible in that it will take big tyres (usually up to 28mm) with mudguards and will also take a rack if you need one. Any versatile roadbike with decent tyres will be fine on gentle offroad. I've ridden my Kinesis from Manchester to Liverpool and back on the Trans Pennine trail (primarily smooth trail, but with lots of stony and rooty bits, not to mention glassy bits where it cuts through estates outside Runcorn) with 25mm Conti Gatorskins without a puncture.
2) Depends on what you mean by budget. Certainly you want to steer clear from anything with the workds Apollo on it . I'd recommend having a look at Kinesis. Excellent aluminium frames. Complete build with Tiagra x10 is about £1,100, or you can buy frame and forks for about £380 and then get the (old) full build kit off Chain Reaction for £330. This is what I did. It's 9 speed Tiagra, but in truth, unless you're planning to do ridiculous miles and/or ridiculous hills it's as many gears as you will ever need. The Kinesis frames are that good that you can then up the spec over time.
3) Nope. But over time you will come to realise that the aerodynamic benefits of not lugging baggy shorts/top round with you are sufficient to help you get over the embarrassment. By that time you will hopefully have lost enough weight that you won't look like a brightly coloured whale on two wheels (not that I'm suggesting for a minute that you need to lose weight like )
4) Good mate of mine who thought nothing of doing the Mary Townley loop on his own for fun on the Saturday before hitting the hills on Sunday for some downhill action - every weekend - bought a roadbike for commuting/fitness purposes. He still mountain bikes a lot but is just as likely to be swooping around Yorkshire on the road bike. In fact I should probably put you guys in touch! Fact of the matter is that road biking has its own magic. When you're gliding along fast on a smooth country lane and powering up (in my case small) climbs, prior to going hell for leather down the other side, it is both peaceful and exhilarating. As mentioned above, there is also something quite zen about mile after mile of tarmac flowing smoothly underneath you. Also, because of the constant pedalling nature of roadbiking (unlike stop/start of lots of mountain biking) as long as you're going at a reasonable lick, weight will drop off in no time. If I had to choose one I'd choose mountain biking for sure, but it'd be a pretty close call..
5) The more you ride a road bike, the more beer you can drink without it having a negative impact on your wasitline I see it as a win/win scenario..