I live in Thailand and have a large dog we brought over from the UK with us. I'm also a teacher!
We did have to shop around for accommodation as some landlords won't allow pets, some buildings don't allow them regardless of the landlord, and some have restrictions on size. But, it is doable and I've now lived in 3 different houses.
Bear in mind that the locations you mention are largely condo living - would your dog be ok with that? Many are, and to be honest mine probably would be now she's an old lady, but I really wanted a house with a garden. This means living further from school but it's fine.
Asia can be very dog-friendly in some regards, e.g. you see them in malls and cafes all the time here! But they are in pushchairs 😂😂😂 Small breeds are very common, medium and larger breeds much less so but some are 'trendy'... Right now in Thailand the usual Pomeranians, pugs, French bulldogs, chihuahuas etc are still very popular but so are beagles, labs/retrievers, and sadly husky-type breeds that really struggle in the weather.
I'm sure there are dog parks in the places you mention, you would just need to research carefully where to go. I imagine SG is very strict on dogs but people must make it work. I'm Bangkok, there are dog walkers and people who host dogs while you travel (there are also traditional kennels but I much prefer leaving mine in someone's home) so I imagine they have similar elsewhere. I rarely have to pay to leave my dog somewhere as I have a nanny, and unless she travels with us she minds the dog while we're on holiday. You will always find helpers to do this.
To stereotype wildly, many locals here are scared of big dogs. I have to tie mine up if I have any workmen come into my house as there is no way they would come in once they'd seen her!!
Nearer the time, once you're looking at specific locations, you can ask in local expat groups for more advice. If you don't have Facebook - get it! Facebook groups are an incredible source of info when relocating.
Best of luck!