Updates are checked daily, with that modification it should not bother you again.
On admin's dashboard we fetch blog/twitter information, but not on the rest of the admin.
Install new locations required internet connection to do it from the admin, but you will not need it if you download the SQL file directly from :
http://geo.osclass.org/downloads/ and then on the admin > tools > import , import it.
reCAPTCHA needs internet connection (that's the way the service is made)
I'm not sure if something else is linked to internet connection, maybe a reference to JQuery libraries, but unless i missed some places... they link to local versions.
If you use firefox, install firebug extension, on chrome open developer console. There're a "net / network" tab on both of them. Reload problematic page, and there should appear all .js, images, css files,... that are called. Identify which one is failing. If everything there, looks normal, then it's a problem of PHP (some files are called within PHP), please tell us which pages are failing for you.
Thanks