Can I save a JSON stream object to Dropbox file with Dropbox Post Rest API ?

To use the Core API, you'll need to register a "Dropbox API app", not a "Drop-ins app":

https://www.dropbox.com/developers/apps/create

Drop-ins don't have app secrets, unlike Dropbox API apps.

And yes, to use the Core API, you'll need to have the user authorize the app to access their account. Then, you can make API calls using the resulting access tokens. (If you only need to connect to your own account, you just need to do this once. You can also easily get an OAuth 2 access token for your own account as shown here: https://blogs.dropbox.com/developers/2014/05/generate-an-access-token-for-your-own-account/ )

Note that getting an access token is a process that involves user intervention, but once you have it, you can store and re-use it for that account without further manual user intervention.

We recommend using an SDK or library as I mentioned before, but if you can't for whatever reason, you can sign the calls using OAuth 1 or OAuth 2. We have blog posts for each:

OAuth 1: https://blogs.dropbox.com/developers/2012/07/using-oauth-1-0-with-the-plaintext-signature-method/
OAuth 2: https://blogs.dropbox.com/developers/2013/07/using-oauth-2-0-with-the-core-api/