1. Personality files are stored like backup images, so they will only replicate to the site server that captured the personality. If you don't like this behavior, replicate it to a UNC path that the client has access to and which is linked to a package (using UNC source) that replicates to all site servers. The reason for this design is that a computer captured in the Chicago site is probably going to be replaced and deployed in the Chicago site -- not Tokyo. If you change this so that personalities replicate everywhere, you may need to be more proactivate about managing disk space. Anyone can accidentally capture a very large personality or personalities which could exhaust disk space on site servers, causing site server unavailability.
2. Yes, check out the PC Transplant documentation for details:
http://www.symantec.com/business/support/index?page=landing&key=55294
3. If you use a Run Script task, and copy files locally, and execute from the client, then it is client side. But since the files for PCT are available on the server, I always run them from the server -- I've never run into a reason to copy them locally.