1.) With converting a thick disk to a thin disk, all the used (non-zeroed) blocks will consume disk space in the target. You could use e.g. sdelete to zero out unused disk space from within the guest OS prior to migrating the VM to reduce the space consumption for the resulting thin provisioned virtual disk.
2.) Once you run out of disk space for any of the VM's virtual disks. ESXi will pause the VM and give you an option to either free up disk space and then resume the VM, or power off the VM.
André