Regarding the airlines, I suggest you find out right now about which one will allow you to have the cat in the cabin with you. The carrier/kennel that your cat will be in, must fit under the seat in front of you (so, that leaves out the aisle seat, right?). United used to allow cats in the cabin, but not Singapore Airlines. Our cat was too big to fit in a kennel that fit under a seat, so he had to be in a compartment down below. It is not the same compartment as the luggage, it is temperature controlled, etc. I am not sure which way would be the best for the animal, I am sure they are going to be frightened/concerned either way. Once you get to Singapore and are at the Quarantine: definitely the air con for the cats. You can go to the website and see what the cages look like (they are quite big) and there will be a number of cages/cats along with yours. In fact, the cats are so close that they can touch noses (which kind of defeats the purpose of the quarantine, doesn't it?). We did not have a "pet mover" handle our cat...we did it ourselves. We wanted to see our cat after he arrived...we didn't want to wait until he arrived at the Quarantine Station. THAT was difficult...we had to hire a taxi (flat rate for 4 hours)...had to drive clear around the airport and to the terminals where they are bringing in import goods (can't think of the name of it), but had to wait in a line to get a pass to enter this area, then had to find the building where the quarantine at the airport is (different from the quarantine station)...the airport quarantine is where they will keep the animal for X amount of hours to make sure it is healthy enough to enter the country. They were very nice, but you have to make sure that someone will be there when you arrive (I suggest that when you show them the paperwork at the airport, you have them call over there and make sure someone is there). We had to wait for over an hour for someone to come back from lunch or whatever...but they were very nice and let us see our cat for however long we wanted. I THINK we had to get the "vet" there to sign off...then we had to go to another place on our way out to pay "tax" on what we imported (the cat)...had to stand in line there, too, but the guy just signed off and didn't collect any money. Then, had to stand in line to return the pass we had originally gotten to enter.
We were told our cat would be at the Quarantine Station later that day and at that time visiting hours were 2-4pm and we went to see him...and he was not in the air con cattery...he was still in the "holding area" in a horrible wire cage, nothing solid under him...he was cramped and HOT, had no water!!!...he was very panicky, it was awful and we were furious! It appears that we had not signed off that he was to have a flea treatment (thought he didn't need the added trauma)... and come to find out, all it is is the Advantage where they put the liquid on the back of their neck. And, in order for him to enter the air con cattery, he had to have this..now, if they had told us this ahead of time, we would have okayed it, since we were paying for the air con cattery right from the start...but they never said that it was a "must" for all cats that would be entering the air con. Anyway, once we okayed that they put him into the air con cattery. The employees were nice at the station...their visiting hours sucked (at that time it was 2-4pm everyday, but closed on Sundays) and they are located at the ends of the earth! Someone has mentioned that they have changed their hours...I hope for the better!
I suggest you bring your own cat's food with you, so they don't have to change what they are used to. Because, unless you feed your cat the same thing the station feeds the cats, the cat will have to make that change, too, and some cats get very upset about that. Of course, the station will charge you extra for the privilege (yes, they charge YOU more for providing your own food).
I must say I met some of the nicest people at quarantine...after all, you have a lot in common...you have a cat(s), you are in Singapore for possibly the first time, you had to get yourself to the quarantine station either by car, bus, foot...only to be ignored by your cat after 10-20 minutes (after it took you an hour+ to get there!)...cat owners know all about this, right?
On hindsight, I think I would go with a "petmover" if I had to do it again...I think they will let you see your pet there at the passenger terminal, before they send it across the airport to the airport quarantine. At least, this is what I have been told. Every airline is different, so you would have to check.
You have to check out EVERYTHING and ask questions (of course, if you don't know the right questions to ask, that is where the problems begin...)
Hope this has given you some insight and was helpful.
Good luck with your cat and your move to Singapore!