Web application + Mobile App Python/Django architecture
I don't use Django but I am currently using Angular with Flask as my REST backend. I think the hybrid approach is useful if your app benefits from caching rendered content. An example would be something like a blogging site where you may store Markdown but render HTML and thus the content is largely unchanging. If this content is shared across many users it can be very beneficial to generate once on the server and serve it to many clients.
However if your reasoning is that because mobiles are relatively under-powered server-side rendering will improve the render time, this does not necessarily hold true. Serving the JSON data model will typically result in less network traffic (obviously dependent on browser caching) which is important for mobile client responsiveness.
For my own part I prefer to render on the client and keep the server-side rendering to some very minimal templates. I find this results in a much better REST API that I can then use to provide external developer access or plug in other client UIs. My client code lives on the client side in Angular, my REST interface is clean, and security is always uppermost in my mind because I have to assume that the client code is potentially hostile.
That said I can see a use for server-side rendering even with Angular. For content that is fairly static it would be great to generate this server side, cache it, and then ngInclude it.
I'm starting a new side project in order to learn some new technologies and I have several doubts about the architecture I should use. My idea is developing an app both for web and for mobile (Android app mainly), so I think I need to implement the following:

REST API service (with django-rest-framework).
Web application (with django).
Android app.

After researching over stack overflow and Internet I have found 3 main "architectures" for the web part (server-rendering, client-rendering, and an hybrid model). In first place my idea was using the client-rendering model (with backbone or angular for the web client). However, I have seen that the hybrid model is a better choice.
So here are my doubts:
1) Could I use django for server-rendering the web application pages?
2) Does Django web app "use" the API or the API is implemented using the same "library" as the django web?
3) Could/should I use a client framework (angular, backbone) for the web client (being server-rendered) ?
Thanks in advance, and sorry about my short knowledge about django, etc.. This side project is just for that, to improve my knowledge about all this stuff.