As to Mary sopping up Jesus’ blood, as I understand it, this is a practice done by pious Jews even today. The rationale being that the blood is as much a part of the body as any other and so ought to be buried with the body.
Yes, this is true. There was a documentary on the Discovery Channel back in 1998 or 1999 about the Shroud of Turin that explained this custom.
But, there doesn’t seem to be any conclusive evidence of this being done by Mary or anyone else in Jesus’ case in the Gospel accounts.
According to tradition, the Sudarium of Oviedu was the other cloth that may have been used to sop up the blood on the face of Jesus. However, we do not know if this was used to sop up the blood at the scourging. This account is found in Blessed Katherine Emmerich’s Dolorous Passion:
When Jesus fell down at the foot of the pillar, after the flagellation, I saw Claudia Procles, the wife of Pilate, send some large pieces of linen to the Mother of God. I know not whether she thought that Jesus would be set free, and that his Mother would then require linen todress his wounds, or whether this compassionate lady was aware of the use which would be made of her present. At the termination of the scourging, Mary came to herself for a time, and saw her Divine Son all torn and mangled, being led away by the archers after the scourging: he wiped his eyes, which were filled with blood, that he might look at his Mother, and she stretched out her hands towards him, and continued to look at the bloody traces of his footsteps. I soon after saw Mary and Magdalen approach the pillar where Jesus had been scourged; the mob were at a distance, and they were partly concealed by the other holy women, and by a few kind-hearted persons who had joined them; they knelt down on the ground near the pillar, and wiped up the sacred blood with the linen which Claudia Procles had sent.