It was normal for a man being crucified to carry his own cross to the place of execution. Apparently, they followed the same procedure for Jesus initially (Jn 19:17). But it would appear that, following his flogging Jesus was already in such a weakened state that he couldn't continue, and it is recorded that, as they led him away another man (concerning whom the writers supply not only his name, but even the names of his sons) was pressed into service (Mt 27:32, Mk 15:21, Lk 23:26).
The task of crucifixion was entrusted to professional Roman soldiers (Mt 27:27-36, Mk 15:16, Lk 23:47, Jn 19:23) for whom crucifixion was a routine task, and failure to carry out such an order was punishable by death.
The Jewish rulers wanted to be quite sure Jesus was dead. They were present to witness the event (Mt 27:41, Mk 15:31, Lk 23:35) and even objected to the wording of Pilate's superscription on the cross (Jn 19:21).
The Roman soldiers wanted to be quite sure Jesus was dead. This was to be a quick execution, since Jewish customs did not allow the bodies to be left hanging on a holy day. However, as noted above, they couldn't afford to bungle the operation. They broke the legs of the two thieves, a standard method of accelerating death. Jesus, who we must remember had been flogged prior to his crucifixion, appeared to be dead already: but that wasn't good enough for them: so one of the soldiers stabbed him in the side with his spear, just to make quite sure. This incident is mentioned only in John's gospel, where it is strongly emphasised that this is not hearsay but an eye-witness report (Jn 19:31-5).
Following Jesus' death, the bodies were not immediately removed from the cross, but were left hanging. Jesus died at about 3pm (Mt 27:45-50, Mk 15:34-7, Lk 23:44-6) yet the bodies were not removed until the evening (Mt 28:57-8, Mk 15:42-6, Lk 23:50-3, Jn 38-42 (this last reference, though not mentioning the time specifically, indicates that Preparation was drawing on and the nearness of the tomb was an important factor, implying it was almost dusk)).
Pilate also wanted to be quite sure Jesus was dead. Mark records that he would not agree to the removal of the body, until after he had summoned the centurion and obtained confirmation that Jesus had been dead for quite some time (Mk 15:42-6).