I have a Clairvoyant staying at my Inn and I placed my catchers on the spots she said they would be, but at the end, when I get the report, it said no one was there for two of the places. Shouldn't teh animals actually be there if the lady says they are? Or is there a chance she is wrong? 

This happened the last time I sent out catchers following the Clairvoyant's animal guesses. She said they were there, but they weren't I didn't think to sceenshot it though. I was prepared this time in case it happened again and it did. 

 In the descriptions, it says pumas ran from both of the empty places, but shouldn't my catchers have a chance to catch them since they were at those places before they ran away? And it says they were empty, then that pumas ran from the empty places. And why didn't my bird catcher interact with the Roc there? He only interacted with the two pumas that came from the empty places. :/

Clairvoyant does exactly what a Cartographer's Guild does. It tell you the current location of the creatures. You have to guess where they will travel next.

But if they are there in the first place, shouldn't my catchers have a chance to get them before they move? And my bird catcher didn't interact witht eh Roc and the Rock didn't move.

Your catchers will be on place only on the next cycle when the pets are already moved to their next locations. You're sending an expedition, not teleportating the catchers to the place instantly :)

Also your catcher can't use the time machine and see what pet was on the certain location at the previous cycle before he arrived, that's why the logs tells you that the location is empty.

Pets move each cycle, they don't stay on the same place for the several cycles. There can't be a situation when the Roc didn't move.