I had a colleague of mine (who works on rockets/spaceflight, has done so for decades) say the barge landing was impossible.
As far as your other demanding questions, no breakthroughs in physics have been necessary to make reusable launch possible. People have constantly made predictions that somehow physics precludes reuse, let alone fast turnaround reuse, from being possible. But that has never, ever been the case. Still people (who ought to know better) will continue making such ridiculous statements. And yes, SpaceX has disproven a lot of these myths.
Privately developed orbital rocket (i.e. not based on government missile parts, etc): Falcon 1, done.
Privately developed capsule return from orbit: Dragon, done.
Vertical landing recovery of a first stage for an orbital rocket: Falcon 9, done.
First landing on an autonomous ship: Falcon 9, again, done.
In less than a month, they'll also do their first orbital flight on a reused first stage.
Lesser-known accomplishments by SpaceX:
Merlin 1D has a thrust to weight ratio of about 180-200 in its latest incarnation. That's dramatically better than the next-best, the NK-33 (at 137). And it's a big reason they've been successful at recovering so many stages without a prohibitive performance penalty. And it accomplished this by using deep propellant subcooling, which had never been done before except in some subscale tests.
Supersonic retropropulsion of the first stage is also something that had never been done before. It's a key enabling technology for Mars landings, and NASA was scared to baseline it for any of their Mars architectures and couldn't afford the tens of millions for a dedicated test. SpaceX gave them the data for free (in exchange for some in-kind services like data collection, but no money). Now every NASA human-scale Mars lander concept assumes the use of supersonic retropropulsion. This is a big change.
Crew around the Moon is not a PR stunt but a revenue source for SpaceX. Red Dragon is not a PR stunt but a key in SpaceX's (and the world's) ability to land large payloads on Mars and test the ISRU technology needed for sustainable crewed Mars missions.
Raptor, which has already had some test fires, is a truly scifi engine. It'll be the highest performance engine (in terms of chamber pressure, which is a key metric for thrust to weight ratio, and especially Isp for a given propellant combination, and also for thrust from a given engine size and thus for manufacturing efficiency) ever. A full-flow staged combustion engine, two-stage turbopump. It's insanely high performance and will get similarly high thrust-to-weight ratio as Merlin in spite of using a less dense propellant.
The ITS will dwarf all other launch vehicles in terms of raw size, raw thrust, raw payload, and especially its fully reusable upper stage/spacecraft combo.
And it's not even that SpaceX are the first ones to imagine these technologies (although some details are novel), it's that they're actually doing them. And that's a big difference. There are virtually no good ideas in aerospace that were not at least imagined by the end of the 1960s, but the vast, vast majority have languished. There have been several other abortive attempts to develop a low-cost private launch company, and (beside Orbital, which has a very different company mindset now) SpaceX is essentially the first successful one.
There will be others, but they were catalyzed by SpaceX. SpaceX's high churn rate has fertilized the whole American rocket industry. And this is very good. Even if SpaceX fails now, Blue Origin will succeed where SpaceX has prepared the ground. (Blue Origin essentially /cannot/ fail, since it's bankrolled by Jeff Bezos, who has more money than God.)