Hi anxious101,
I'm a Singaporean, so I guess I'll try to answer your questions. (I wont be trying to chase you off too, since I've a place in Duke-NUS, so what I'll say is my honest opinion.)
Yes, Sg housing and cars are expensive, this is due to the limited landspace and the rapid increase in population (and also market inflation and stuff). A typical Singaporean would use a large majority of their income to pay off housing and car loans.
That said, everything else is relatively much cheaper. Food is good and cheap (unless you are constantly eating high-end restaurants), public transport is also good and cheap too, especially since you would have the student pass. (although I still hate travelling during peak hours)
Cars are also considered more of a luxury than a necessity in Singapore, especially so for students. Not everyone owns one.
There are however, many other pros for staying in Sg. It is a very safe country, clean and neat. Majority of us speaks English, so its not difficult to assimilate. Although we are known to be a 'strict' country since we implement fines for many things, it is perhaps, over-rated. If you are a normal law-abiding person who doesn't go out to break rules, you'll be alright here.
Working as an early doctor is tough, but this is perhaps the same with other Sg new grads in their first jobs. In Singapore, competition is stiff (as I said, growing population), but if you are willing to work hard, I'm sure you'll pull through. No one said becoming a doctor was easy. One of the reasons doctors are well respected here is that the studying and training is tough.
I guess being a doctor would allow you to get a decent house and car in Singapore, but if a high income is your goal, my advice is for you to go into the finance sector? Maybe you could earn more money there.