There are several instances of Overpass API that keep up to date with the main OSM database (usually seconds or a couple minutes behind).
Of course exported data will be from the time of export.
No, it won't, unless you manually update it. I am working on making an API for developers, where you can create/update map data programmatically, so you will be able to do it in a cron job for example.
Thank you; that'd be cool if it would, but it'd definitely need to be optional... Don't want some rogue edits making your map suddenly look like crap.