Hi Maurits,
Just some of my personal opinion to some of the questions you have posted. Views and preferences from others contributors may differ but would get you a more holistic approach to your move here.
Housing:
Check out property sites such as iproperty.com.sg or sg-house.com. no harm dropping an email with your requirements and see what throws up. Agree that you should make temporary lodging arrangements so that you can suss things out first. I had a friend once stay at Metropolitan YMCA (@ Stevens Road, a couple of bus stop from Somerset MRT station) throughout the 3 months he was here for his internship even though it was meant to be just temporary.
If you're looking at areas to stay at, those nearer Orchard Road and Somerset are generally more expensive than the suburbs. but suburbs these days have good transportation networks and foreigners from everywhere, which makes opportunities for shared housing. Reckon that areas such as Holland Village/Road, Ghim Moh, Tiong Bahru, Toa Payoh, Bedok may fit your budget. You can even go further out east to Simei or west Bt Batok. They may be further away but considerably cheaper and if you get a unit next to the MRT stations, than you're set for life.
Transportation:
if you're taking a train/bus and just heading to work and back, it's probably about $5-6/day. taxis are way too affordable and accessible here, which makes it deceiving. a third of my salary generally goes to cab companies
Taxes:
I don't think you have to worry about that since i don't think you'll be taxed at all.
Cost of Living:
Standard of living in Singapore is generally quite high.
For a household of 2, i generally spend about $70/week on groceries which usually includes a 6-pack somewhere. but $70/week means i eat breakfast everyday, junk food and snacks and dinner abt 3 times a week for 2. when eating at hawker centres, expect to spend about $6-7/meal and about $10 at food courts. Macdonalds cost about $7/meals. I reckon you'll be quite comfortable if you eat at home or at hawker centres on most weak days and splurge on the weekends. If you're eating at a cafe, the price ranges from $30-40/meal with an alcoholic beverage. Expect to pay twice the amount at a more fancy restaurants.
Social Activities
I suppose you'll meet people at work who will introduce you to their friends and their friends friends. That's what i've noticed amongst my friends who are from overseas. I've friends who go wakeboarding by themselves at first but make friends from that circuit. that's possible but wakeboarding may not be too cheap.
I think you'll have a good time from just socialising with new people, discovering a new place and trying local cuisine and what Singapore has to offer first.
As to travelling around the region. It may be a little tight if you depend solely on your salary. I'm sure you'll be able to squeeze in Malaysia, Bali and Bintan.. but otherwise there's always time to come back later on.