Is it possible to login to Dropbox and authenticate/allow an app to access files in dropbox without a browser?
Like via ajax or curl? I am thinking if there will be no UI.
No, the Dropbox OAuth app authorization flow requires the user to visit the Dropbox site in a browser to authorize the app. That only needs to be done once per user though, after which the app can store and re-use the resulting access token.
Is this the same for all SDKs?
Yes, this is a part of the API and applies to all of the SDKs.