I'm going to be away from my pc for quite a long time and I'm going to be able to use my phone instead, and I'm a fan of the site.
So as stated above, does my day count if I'm connected through my android app?
Just being logged in is not enough, you have to perform other activities.
We're now counting access to a site via the app for both "last seen" and the "daily site access" for the Enthusiast and Fanatic badges.
As is true with the sites, we don't actually count all access; only those actions we consider "real". For the mobile app, this basically means accessing the feed doesn't count; we'll probably tweak what we consider "real" over time.