1) ARM is some kind of open-source processor architecture right? It seems like everybody and nobody owns ARM. From my understanding, the ARM company just engineers the cores/architecture and licenses that out to other companies to fab. They kind of act like the processor R&D team at AMD or Intel for example. You can "buy" their cores to use on your SoC, or you can "buy" their architecture as a building block to engineer your own optimized cores.
Except for your first bit about it being open source and nobody owning it, that's essentially correct. The ARM people own the IP and license it out, either as complete core designs or as a license to use the ARM ISA.
2) Why/how did ARM gain such a strong foothold? Was it just because they were the first to make it to the lowest power usage arena? I have a hard time believing that AMD or Intel didn't see the need for this coming a long time ago when smartphones first started emerging and didn't think of something similar.
To a large extent, I think they were just in the right place at the right time. They had the right combination of performance and power usage, at exactly the time when the mobile device market was exploding.
Other lower-power processors were (and are) out there, like the PIC microcontroller line. But these did not have the compute horespower to handle a smartphone. PICs get used a lot in appliance and automotive applications.
The MIPS processor line could've been what ARM is today, but back in the '90s they got acquired by former workstation/server vendor (and inventor of OpenGL) SGI, and subsequently spun back off again a few years later when SGI made the ill-fated decision to bet the farm on Itanium. IMO this little detour derailed any chances MIPS may have had of dominating the embedded market. They're still around (used in some consumer electronics devices like Blu-Ray players, set top boxes, and the PSP), but haven't managed to achieve the dominance that ARM has.
3) What's preventing others (AMD/Intel) from beating them? Is it just a big black box that nobody can reverse engineer? I have a hard time believing that since you can license their actual architecture like Apple did with the A6.
Beating who? ARM Holdings? They're not a semiconductor company, they are an IP licensing operation. They don't build actual chips, so they don't compete directly with Intel or AMD.
Coming up with a different (but equivalent from a performance per watt standpoint) RISC CPU design that doesn't use any ARM IP is certainly doable (especially for someone with deep pockets like Intel), but there's also a huge existing ecosystem for ARM development. Compilers, OSes, and APIs (Linux, Android, etc.) all exist today. If you rolled a new design from scratch you'd have to port or re-invent all of the support infrastructure too.
Doing it with x86 (to leverage the existing x86 ecosystem) is difficult, because x86 is a complicated ISA with a lot of excess baggage that isn't needed for mobile devices. Atom was Intel's attempt at this, but it was still too power hungry for the sort of applications ARM targets, and too wimpy for low-end laptops and netbooks.