Should I Send the Thief to Pillage?

It is best not to send him to pillage Lothric Castle until you have kill Aldrich and Yhorm, or you will not be able to enter the Grand Archives to retrieve his ashes. If you tell him no, he will not ask again, but there is a menu choice to send him there whenever you see fit.

Should you let Greirat pillage?

It's safe to let him pillage the Undead Settlement area. Speak with Greirat again when you reach the Irithyll of the Boreal Valley area and he will ask to leave again for more pillaging. While you can let him pillage this area, he will not survive the trip unless one of two tasks have been completed.

How do I make sure the Greirat survives?

port back to high lord wolnirs bonfire, go out towards irythrall, cross the bridge, kill the nice doggy, and entry irythrall. activate the first bonfire right in front of you. head back to firelink, send greirat away. talk to patches, if he still has onionbro's armor, otherwise continue.