Memory mapped hardware registers are accessed like RAM. All CPUs have specific instructions for reading/writing RAM and these same instructions are used to access memory-mapped registers. Hardware registers in general do not have to be memory mapped, this is simply a common convention.

There are 2 main ways that software (device drivers) can access a device's internal registers - by mapping those registers into the CPUs physical address space (the same way that memory is mapped into the CPUs physical address space); and by providing a dedicated mechanism (e.g. IO ports on 80x86).

Memory-mapped I/O (MMIO) and port-mapped I/O (PMIO) are two complementary methods of performing input/output (I/O) between the central processing unit (CPU) and peripheral devices in a computer (often mediating access via chipset).

No, those registers are inside the actual CPU (or CPU core for multi-core CPUs). You can not access them through loads or stores to any memory address. A memory-mapped register is something which you access through an address or a pointer (in languages that have pointers).

DMA is when two devices that aren't the CPU use the memory bus to communicate (with one device usually being main memory, and the process being orchestrated by the CPU). Memory-mapped IO is the CPU talking to device on the memory bus that is not main memory.