Firstly, the mobile app is a question that's been asked several times over in several threads within this section. Currently no app exists. Will there ever be one? Can't say. I'm no programmer, though this desire has been noted many times over.
As for the notifications, if you subscribe to threads via email notification (look at the bottom of the page of any thread once you are in the thread, you'll see the link to do this on the bottom left), you will receive emails letting you know whether or not you are logged in. You can do the same for your PMs. The trick here, however, is that you need to have your emails on. So, it'll have to be to an email address you check often if you want to make sure you get the updates as they happen. But these emails will come to you whether or not you are logged in.
Anyway, hope this helps.