Did Antony shake the conspirators bloody hands?

This is true. Anthony did this because of his curiosity. He wanted to know what made the conspirators do what they have done. He also shook hands with them in order to know that they are guilty and to also remember details about them. After this, he then started to make an effort to give Caesar a proper burial.

The whole time that he was doing these things, he was already planning how he is going to avenge Caesar’s death. By knowing the plan, he knew the right time when he would strike so that he would make the most out of the time that he has.