Hi Chris, an aussie here.
I"ve done a few trips now. In Asia, only two long ones in india, both on a moutain bike but there most roads are paved and a bike with bigger wheels is fine. The more spokes the better though.
I'm on my third bike now. I'm sticking with Giant as the best value for money bikes you can buy although i'm not saying its a fact but its my impression. Reason being that they make so many they get good economies of scale. And the quality is excellent.
I'm on my second giant. The first one was stolen in Paris. I bought my second one there and its got bigger wheels. Bigger wheels enable you to go faster and you can ride them on rough roads anyway. The only reason i'd suggest you buy a mountain bike is if you are really heavy and intend carrying a lot of gear. Personally i suspect you don't need a mountain bike for your asia tour and you will be better served with a touring style for Europe.
I'd buy a bike that would serve both tours to save you more money. And one decent Giant bike will certainly be adequate for both tours.
I bought mine in Paris it was the cheapest touring bike i could buy and it came fully rigged with lights, mud guards. I recommend you buy better peddles and put on either spd clips and buy the shoes or a cheaper option is to buy a cage or toe clip type gizmo. Either way, you should get one or the other for efficieny and comfort.
The seat on my bike is great. The bike cost about $600 at a time when the exchange was good. Only in 2013. If you do not have a lot of money, i am all for buying the cheapest bike that will do the job reasonably well because you will have to buy more gear and should keep most of your money for enjoying your trip.
Buy ortleib panniers. For Asia, you should be able to make do with only the back panniers as you won't carry a tent most likely.
As for tyres. You probably don't need knobbly tyres unless you plan to do a lot of offroading. I would stick with slicks of a wider kind.
Remember when kitting up, including with tools, less is generally more. Asians ride bikes so they have bike mechanics everywhere except perhaps in more mountainous areas but they should be able to help you with any mechanical problems.
Another reason for buying a new bike is that you are unlikely to have any mechanical problems. If you buy a second hand bike you will have to get it serviced and who knows what issues will arise.
With any cheaper end bike, ensure you have a "sealed bottom bracket". If the bike does not come with this, get it changed over before you leave. This brought me unstuck towards the end of my tour in outback australia with my first Giant. The gears seized up and I couldn't ride another inch. And no one locally could fix it.
Only get suspension in the front forks, not elsewhere on the bike. Sorry I can't remember what model i bought. I would also consider the advice of people in the bike shops when choosing one.
I replaced the pedals on mine with the steel ones I had on my very first bike which I found in the lifeline shop. If you buy second hand, make sure its the right size and get it checked out first by a bike mechanic so that you know how much extra you are up for.