You certainly can run DOS on modern x86-based PCs. There's a few issues though with modern hardware being somewhat incompatible with native-dos games(sound card, mouse...). Also, some games will crash or malfunction if the CPU speed is too fast, or there's too much RAM. The reasons for emulation/virtualisation are usually to solve the incompatible hardware problems, and/or to enable software written for another O/S or system to run at the same time as your main O/S.
Other emulation/virtualisation software has it's pros and its cons. Virtual PC and VMWare (from my experience) offer more complete "emulation" but don't support as many old hardware devices, and they tend to be slower. Not sure about DosEmu as I've never used it, but I think it's more geared towards speed than compatibility, whereas dosbox is more geared towards compatibility than speed.
Triple, quadruple...you can have pretty much as many operating systems installed and bootable on your computer as you like.
I think the 64-bit CPUs can't run older 16-bit x86 code, so they're not strictly x86 compatible. Otherwise, yes. These days, most people would consider a PC to mean an x86-based computer.
Not sure about the slowdown question. Emulation is more accurate...