To build on what Bastien said, in terms of if you’re wanting to use existing integrations and don’t have the resources or desire to write your own…
External apps which integrate with Asana basically all work by using the Asana API to read (and where relevant) write to Asana’s data. That API work occurs “in the background”; i.e. it’s not user-interface-related.
On top of the API communications, external integrations typically need some type of user interface to perform the user-interaction part of their integration. That UI aspect can be implemented in a number of ways - it could be done via a Chrome extension, via a separate website or web application, or an installable Windows/Mac application, to name three common approaches.
With that background, let’s tackle your questions…
No, as mentioned above, the user-interface part of an integration MAY work only on a browser if that’s how the external app developer chose to implement the UI portion; but it may not require a browser if they chose to write a separate app, etc.
It definitely could, if the developer wrote it that way. As mentioned the Asana API itself is always behind-the-scenes; it has no UI elements to it.
If you’re just talking about using an existing integration, it’s nothing you can set up; the developer has to have provided a way to do that as part of their integration.
If that’s the case, then it sounds like they’ve all chosen to implement the user-interface portion of their integration via a Chrome extension. If that’s true, then unfortunately you’re out of luck in terms of using them with a mobile-only solution.
I know my post here was a long-winded way of getting to this answer of “you’re out of luck”, but I wanted you to understand all of the background of the answer, since you asked to be “unconfused”.