All the Apple //c machines from September 1986 onward have an additional 34-pin header on the motherboard designed for memory expansion boards. Apple offered their own expansion board which support up to 1MiB of additional memory internal to the Apple //c. Other manufacturers, such as Applied Engineering offered 3rd party versions of these memory expansion boards, such as the Z-RAM Ultra that you mention. Also, the Laser 128 and later Apple //c Plus supported their own slight variations.
The 34-pin header includes many, but not all of the 65C02 bus signals. The connector is really purpose-built for these memory expansion boards, and there is firmware in the ROM of the Apple //c that allows them to be used as RAM disk devices under DOS, ProDOS, or Pascal. The memory expansion occupies Slot #4, from the viewpoint of the OS or application software, and supports the SmartPort device protocol to look like a disk ("block") device. It is software compatible with RAM disk expansion cards designed for the //e as well. The normal configuration under ProDOS would allow you to access a new volume called "RAM4" as a RAM disk. See this answer for some more info on access to Apple //c "Slinky" memory expansion.
Since this motherboard connector is connected to the 65C02, other peripherals, such as the mentioned real-time clock are certainly possible. Also, since you are adding a circuit board with memory and firmware, adding additional capabilities like a Z80 to run CP/M also become possible. I suspect that for the Z-RAM Ultra, the Z80 "takes over" the expansion memory and uses it as local memory, while also using its interface to the Apple //c bus to access the other peripherals.
The motherboard connector is not physically like a slot on an Apple //e. The signals available are much more limited. So, I think that 3rd party developers got creative with how they added features to these boards and likely developed proprietary software methods for communication between their expansion board and the rest of the Apple system. The connector is documented in the Apple //c Technical Reference manual, but the information is limited to just the basic schematic and pin-out. I think that 3rd party developers began by just reverse-engineering the Apple memory expansion card to some degree, and later added the multi-function enhancements to make their product more competitive with Apple's.
Update: As pointed out in the comments, the Z-RAM Ultra (specifically asked about by OP) bypasses the memory expansion connector and fits into the IC sockets of the CPU/MMU on the Apple //c motherboard. Thus, it would not be nearly as limited on controlling the 65C02 and everything else connected to it, as this makes it more of a "superset" of the signals available from a regular Apple //e slot and not a subset of those signals.