Download Blob Container or Folder From Managed Disk And Create Zip C#

Hi,
Currently we are working on a requirement where client wants to download selected files from Azure Blob or Managed disk in a zip folder.
Background :
1. There are two types storage that clients are using it can be either Azure Blob or Managed disk.
2. The files will be selected by users can be in one or different folders or containers on respective storage type.

Requirement :
1. Collect selected files from different locations on storage server i.e. Managed Disk or Azure Blob
2. Create a Zip file and download it on browser
3. Store that zip file on respective storage for keeping log of download and provide re-download option in download history

Approach we have adopt :
1. We are copying all selected files in new folder from specified path on respective storage types itself (As we don't want download files on app server and we want to maintain log that what files are downloaded by user)
Challenge we are facing :
1. Is there anyway that we can zip the newly created folder on storage server itself and download? If not then what will be the best possible way to do so?
2. Is there anyway that we can download entire folder(managed disk) or container (blob) and zip them at code level in c#?

Suggestion:
1. If above approach is not up to the mark then what will be the best approach using Asp.net?