Hello,
I'm an experienced developer. I have used Google apps for my personal projects and also I have done Google apps projects here on freelancer, my last 2 projects are Google apps projects.
I have gone trough your project details and this is how i would finish your solution from reading your description.
If you don't have them I would create spreadsheets for hospital and worker schedulers based on your master spreadsheet or your design.
1. Part
I would use onEdit spreadsheet trigger to save the changes in a hidden sheet. When the user is done editing he would click a button somewhere on the ui, most likely sidebar what would push the changes to you master sheet. This can be done in many ways but from reading about it it seems the best way. You can set the edit/view permissions manually.
2. Part
This part seems easy since it's a one way sync. You can set the view permissions manually or create a viewable link.
3. Part
Hospital and worker schedulers would be in there separate folders, I would sync with all files in a folder. I would probably detect edits the same way as in part 1 but the sidebar would have separate controls for hospital and worker sync.
This isn't for certain but just a general idea since don't know how your master schedule looks like and if you don't have the spreadsheets already made I would like to have some kind of a design.
Feel free to contact me if you have any questions
Regards,
Ognjen Jaric