Unfortunately correct until SE adds write support to the API, which we're told is in the works. For now, we provide a simple way to open questions in Mobile Safari which of course doesn't have this restriction.
It's currently in private beta and should get submitted to the app store in the next couple weeks. This will be released as an update, so current users will gain access to the full SE network. :)
You provided great arguments for 1 and 2. Adding a "my questions" view and an "all my tags" view are certainly worth considering and wouldn't be difficult to implement. Number 3 is actually implemented already. Once you select a tag or view all questions, pull the detail view down to reveal search and sort controls. We're considering showing these controls by default as users are not finding them.
Thank you for the great feedback and we're glad you're enjoying consuming SO content in a new way!
EDIT: Also, as soon as the Stack Exchange API supports write access, we'll be all over that.
http://bigbigbomb.tumblr.com/post/17252346935/learning-from-...