I'm sure that Dave will confirm but my understanding is that DF20/DF21 are only transmitted in response to the UF20/UF21.
UF20 being a request for altitude
UF21 being a request for airframe ID
Yes - UF20 and UF21 are the extended versions of the standard UF4/UF5 uplinks, which request altitude and squawk respectively.
The difference is that the UF20/UF21 also contains a request for the contents of one of the transponder registers, which are sent in the DF20/DF21 response.
The registers normally requested are:
for ELS:
BDS1,0: Data Link Capability report
BDS2,0: Aircraft Identification (i.e. Flight ID)
BDS3,0: Active TCAS RA(s)
and, for EHS:
BDS4,0: Selected Vertical Intention
BDS5,0: Track and Turn report
BDS6,0: Heading and Speed report