I went to singapore to see my parents in march with a 24 month old ds and a 5 month old dd and was panicking about all the same things!
In the end we took a night flight out which was fine, dd slept really well in the sky cot/ bassinet thing - definitely book one of those if possible. Other advice I'd give about flights is to book a seat for 20 month old. Having her on your lap will be HELL!!
Also, try and book your dh and dd in a different row because the bassinet is on the wall in front of the saets so the tray and tv screen come out of the armrests which don't lift up. My ds found it really hard to sleep sitting up and if he could have laid across 2 seats he would have been more comfortable.
I decided in the end not to take steriliser/ nappies/ wipes/ food etc. I took one of those microwaveable steriliser bags for the teats and just made sure I washed each bottle as soon as she'd finished it.
Nappies and wipes you'll be able to get out there. The only thing I took was hypoallergenic bath stuff for DS as he has eczema and I didn't want him to flare up. Everything else I bought out there and left behind for next year.
I also got my step mum to check out baby food, formulas, etc before we went so your MIL could probably do the same for you. You'll probably be surprised how little you actually need to take. We had more hand luggage than hold luggage!!!
I also took 2 buggies - v useful for carting everything around the airport, and you can take them up to the door of the plane as you board and they put them by the door as you arrive.
I wouldn't worry at all about the weather. My 2 didn't even notice the difference I don't think. The best thing I bought was those UV sun protection suits that come down to their knees and elbows so that there's only forearms and calves that need sun cream on. They were fantastic, they wash easily and dry in no time. My 2 lived in them and hardly wore anything else for the whole 2 weeks.
There are loads of tips in the travel archives on here - I got loads of ideas from there.
Oh, and make sure you take a change of clothes for you and at 2 changes of clothes for the kids and twice as many nappies as you need on the plane. MY dd puked all over me 10 minutes into the flight and I had to sit in wet, stinky clothes for 11 hours - nice! 
Sorry to go so long - good luck and have fun!
Hope some of this helps, don't worry about it too much, I found I was panicking about nothing in the end and had an amazing time.