identity of application pool is found in IIS in application pool properties, I have no way to know what user it is on your machine or hosting environment
yes moving from one machine to another messes up permissions on existing files because each file has an owner user sid and that does not exist on the new machine
folder permissions might be ok because new files get uploaded fine but permissions on existing files are hosed up
for the log file /Data/currentlog.config you can try delete the file and it may come back if folder permission is correct it will be re-created.
Shared Files are uploaded under /Data/Sites/[SiteID]/SharedFiles
you should check permissions on individual files there, since new files work compare permissions vs the folder permissions which seem to be ok since new files upload ok