I'm providing this advice to you on my personal capacity - as a Singaporean who has lived here for over 20 years, and have studied and work in the USA. You should always ask around for more advice, but here is mine - to you - point by point. (Sorry to seem so 'German')
1. It is a common misconception among expats that staying in HDB is not as good - sure it has no facilities and the place may not appear as 'swanky' and you save alot of money. HDB flats are not neccessarily 'boring' - in fact there is more to do - there are HDB that are located near to public facilities - pools, community clubs, playgrounds, parks. And often they are near eating places and very well connected by public transport. At your salary, you are not going to afford staying in a
condo, ok, maybe you could afford a small one - but in the end, you will be left with no savings.
2. Staying in Malaysia (2nd Link) is not a bad idea. There are people who do it. There are many options there and you could get a landed home for say $1,000 Sing dollars (if memory serves me well). Since your wife is Msian, she should have no problem settling there. However, I believe that the commute could 'kill' you. Prepare to devote 2 hrs (one way) travel every day. That's 4 hrs travel a day.
3. Definitely have your wive's relatives along for the viewing. (In fact, you should probably view it 'seperately') And avoid the temptation of getting intimidated/threatened by the agents. They are a crafty bunch. Do your homework, know the market, before you make the offer. Don't do things like give a blank cheque or sign the option without the seller being there.
4. I'm not sure how to advise you on the counter offer. So many variables exist - but I think Singaporeans work on reason and logic, so if you can supplement your asking with a good one, do it. Also (of course), ask with politeness and humility, make it clear that it will not affect you taking or not taking the job. At $5K a year more, it should not matter to them.
5. There are many educational institutes in Singapore, and I think there is a good chance that your wife can get a job with part time teaching in an institute - although that is less flexible and takes up more time. The other option is for your wife to be a student tutor in common school subjects like Math, Science and Economics. This is flexible and can be lucrative, particularly if teaching at the 'A' Level or Uni level. I know a friend's wife who makes $4,000 (sing) a month tutoring. All Singaporeans speak english (many do), doubt there is any money in that.