Unblock the zip file before any contents are extracted. Once the zip file has been unblocked the contents will not need to be unblocked after they have been extracted.
I don't think that there is anything that you can do to prevent "The Mark of the Web" from being applied to downloaded files. For example, even zip files that I have uploaded to my own OneDrive storage will need to be unblocked if I download them! However, since the Mark is actually an NTFS alternate data stream if you save the download to a non-NTFS file system (e.g., USB thumb drive formatted with Fat32) then the downloaded files will not be blocked.