So does it mean at first it checks for updates in server and files in PC and if the previously downloaded update is no longer necessary, it will delete that file but if is necessary it just leaves and downloads the new ones only?
WSUSOU downloads the wsusscn2.cab catalog from MS (the same that WSUS uses). That catalog tells WSUSOU what updates are new, and which ones are no longer required (as they are superseded by newer ones). It will check all updates for valid application signatures by MS. Additionally, all files in the WSUSOU repository will be
checksummed and validated at later runs. To your question: Updates that are not new and not superseded will be re-downloaded only if found corrupt. Otherwise they stay untouched.
Moreover, if I move WSUSoffline to new PC and also copy that client folder, would be redownload all updates or act as if updates had been previously downloaded on the same computer?
If the directory structure is left intact* all present updates will be found and not re-downloaded.
*Please use a transfer method that does not change update timestamps. Transfer methods that might change timestamps in certain include (but are not limited to): FTP/SFTP or FAT-formatted USB drive (FAT drives have a timestamp resolution of 2 seconds, while NTFS has high-precision timestamps).