So far refrigeration related to HTS is concerned...80 K is the economically viable solution to use in case of HTS technology using liquid nitrogen. But when we talk real techno-economic way between HTS capital investment versus operational costs..one should operate the system under cryo environment at less than 50 K bcz there is great benefits of exergy (available useful energy)....even Neon refrigeration at 28 K is also done but very specific, as Neon is not available in large quantity. Other than Neon, pre-cooled helium gas at 50-60 K is ok. 

The unreasonable price of HTSC  makes to use it at 4.2K. However there are much more convinient LTSC conductors for utilisation at this temperatures, if only your aim is not very high field solenoids.

Depends what thermal mass of HTS you are cooling. For small amounts the cooling power from 60 to 80 K Stirling cycle coolers is sufficient. For larger thermal masses liquid nitrogen cooling is viable.