Can I do this with Dropbox API?

This sounds like it should be possible. Specifically:

- There isn't a special kind of account for this, but a normal account should work. You can register an account here: https://www.dropbox.com/register

- Yes, you can use the new API v2 shared folder endpoints to share folders and invite users:

https://www.dropbox.com/developers/documentation/http#documentation-sharing-share_folder
https://www.dropbox.com/developers/documentation/http#documentation-sharing-add_folder_member

Note that these endpoints are still in beta though.

- You can use webhooks to be efficiently notified of when things change in the account: https://www.dropbox.com/developers/webhooks

Moreover though, you probably don't actually need to use a central account with shared folders like this. If you just register an app normally (even just the app folder permission would suffice) and have users link their Dropbox accounts to it, your app could still be notified of changes in their accounts via webhooks, and then use the normal file uploading and downloading calls to get the contents and replace them, delete files, etc.