Do BJT (Darlington) transistors have a minimum collector current?
No, you can't assume that, at least, not in general.

A Darlington transistor pair operating at full gain has a minimum VCE that is a combination of the VCE(SAT) of the first transistor and the VBE of the second transistor. In order to conduct current at all, the overall VCE must be greater than the sum of these two. Any less, and the second transistor does not receive the base current that it needs.

However, with sufficient base drive, the Darlington pair can "revert" to single-transistor operation, in which the first transistor ceases to provide any gain, and simply provides a path through its B-E junction for the device base current to flow directly to the base of the second transistor. In this mode, the VCE of the device overall can be the VCE of the second transistor alone. I think this is what Figure 11 in the datasheet is showing.

But if you have that much base drive, then there's no reason to use a Darlington in the first place. You'd be better off overall using just a single transistor.