Do the mystics say? And when was the Sanhedrin’s illegal trial held? Can pilgrims see the dungeon where Jesus was kept?
Just from the gospels, we can infer that it was held deep in the night. The rooster crowed when Peter denied Jesus; the third of the four Roman night watches, the ‘cockcrow’ (gallicinium / alektorophōnia), was between 12 AM midnight and 3 AM. The final watch was called prōi, ‘early morning’. (cf. Mark 15:1 “Early in the morning (prōi), held a consultation with the elders and scribes and the whole council. And they bound Jesus and led him away and delivered him over to Pilate.”) The rooster could have crowed somewhere around those early morning hours (it does not necessarily have to be literally on the third watch).
As for the place, we can’t really be sure where Caiaphas’ residence exactly is because as in a good number of sites in the Holy Land, you have multiple rival claimants. St. Peter in Gallicantu on Mt. Zion is the more famous site, but the Armenians have another site (also on Mt. Zion) which they claim is the actual house of Caiaphas. The only thing we can be more or less sure is that Caiaphas’ house would have been in the Upper City, where the priests and the aristocrats of Jerusalem lived. (Mt. Zion was part of the Upper City, so both St. Peter in Gallicantu and the Armenian church are at least in the correct area.)
Why was the cock crowing so early in the morning?
Cocks always crow early; they have to wake the sun up.
It was probably near the house of Kathros, which is in the Old City and not far away from the Second Temple. The area of Zion is a ways off, and the Ananus-Caiaphas families were always involved in duties at the Second Temple, so a close location would make sense.
Josephus also places the palace of the Asamoneans (Agrippa’s eventually stayed there) as right across the cardo from what is now the West Wall of the Second Temple.