Even if SpaceX made the launch costs a hundred times cheaper than the shuttle, it would still cost over $500 million and that would only launch a tiny fraction of the mass needed for a full artifical gravity space station. The estimated cost of the International Space Station so far is over $100 billion not including launch costs so even if another company cut costs there by a factor of ten, thats $10 billion, again a small fraction of what you would need. You can cut out bureaucracy and save money with a private company instead of government agencies but this would be a one off design made for space habitation so there really isn't much room to drastically reduce costs. If you want this thing to also travel around the solar system, you'd need to complicate the design even more because the ISS is solar based and has only orbital correction propulsion which is much cheaper.
It's possible but I don't think it will be economical or practical for a long time.
Yes, but the US Navy builds them one at a time, every 5 years, so $2B/year and they are used to guarantee the cooperation of an important ally/partner with the USA instead of with Russia or China. There is a big cost, but there is a monetary benefit from the deals made by promising to always keep a carrier strike group near someone's shores, ready to defend from foreign aggression.
NASA's budget in 2015 was about $18 billion while the defense budget was just shy of $600 billion. If we lived in a fantasy world where NASA was considered as important as defense, we'd be an interplanetary species by now. Alas, we don't, and when we consider the feasibility of an idea we have to take that into account.
That is entirely unclear due to how the geopolitical landscape has been evolving. SpaceX's major innovations amount to cutting out corporate bureaucracy, cutting operating costs, and taking advantage of economies of scale not available to the last generation of aerospace manufacturers. SpaceX hasn't invested in science, they've invested in engineering and logistics that make them competitive with the current state of the art, up to and including their first stage recovery technology. For example, they use kerosene instead of liquid hydrogen, cutting out the huge manufacturing and storage cost, and because they're not subject to Congressional pork barreling, they don't have to split manufacturing across many states and districts. Whether they can make the leap from basic chemical propulsion to something that can really drop costs by an order of magnitude or two remains to be seen.
It's possible but I don't think it will be economical or practical for a long time.