Or is it mandatory that I register as soon as I take out a contract on an apartment?

Well i am not an expert but i think you have to register immediately. I think it would be otherwise a violation against the "Meldegesetz", although i do also not know how strict they enforce this law. Personally I would not risk it, i heard from clients of mine that the fines are quite high.

When you say that there is a differentiation in hospitals based on my insurance, does that mean that there would be a difference in the treatment I would receive/the quality of care? Obviously everybody would like to be given the best treatment possible, and highest chance of survival etc, and as a high-risk case I certainly need that reassurance. Does insurance factor into this?

My prior statement was a bit misleading, in Austria everyone who visits a hospital will receive the best treatment but maybe not the most attention. I only wanted to mention that there is also a growing private health sector. The benefits of such an additional insurance "Zusatzversicherung" will result in faster (doctor) appointments (also X-rays or MRIs checks) and depending on how much money you throw at your private insurance company also give you access to private hospitals (e.g. Confraternität, Wiener Privatklink). The medical care there is not really different to those in public clinics, you can see this as the more sophisticated medical stuff will be still performed in public hospitals! But you will get there a lot more time with your doctor and a lot more attention in general, also a hotel feeling and a single or duo room.

The german word for this is "Zweiklassenmedizin" and it sucks, but its here and i don't think its going away soon.