It’s great to see interest in PWAs for Shopify! PWAs offer offline functionality, faster load times. To answer your questions:
The issue with the site.webmanifest file not saving changes might be caused by caching or the way Shopify handles theme assets. Sometimes, edits to files are not reflected immediately due to cache, so clearing it or making sure the file is correctly placed can help.
For offline capabilities, you will need a service worker, which can be added through your theme.
I found a detailed guide on how to integrate PWAs with Shopify. It goes through the necessary steps, such as configuring the manifest.json file and setting up the service worker. You can explore it for more details: : How to integrate PWAs into Shopify .