Xbox – Does the Bowcaster kill in one shot

From Microsoft's official statement post-the E3 uproar, back on the 19th of June:

An internet connection will not be required to play offline Xbox One games – After a one-time system set-up with a new Xbox One, you can play any disc based game without ever connecting online again. There is no 24 hour connection requirement and you can take your Xbox One anywhere you want and play your games, just like on Xbox 360.

^{(their emphasis)}

That "one-time system setup" includes an initial day one update of about 500MB that the console needs to download to be fully functional, but you don't need to sign in to your console or Xbox Live to receive that update, the console will find and download it as soon as it is first internet connected. (source: this was the first thing my Xbox One did yesterday after switching on and checking my wired internet connection, and before taking me through the initial set up of signing in and calibrating the Kinect).

Of course that statement was published under the name of Don Mattrick, then President of Microsoft's Interactive Entertainment Business, who has since left Microsoft to join Zynga, so I can understand anyone worrying if that statement is still true.

Marc Whitten, who is still, Microsoft's Xbox One chief product officer was interviewed by Kotaku, and this is what was said:

...your new Xbox console would have to connect online once in order to do the things Microsoft described today. And then you can keep it offline and play games without re-connecting to the Internet forever.

I don't believe that there has been anything since contradicting that statement.

(Addendum: it was briefly possible to pull down the day one patch on a PC and transfer to an offline Xbox on a USB stick, but Microsoft have just removed that facility from their website, and have said that Microsoft Support will take people through a manual update process if necessary)

Lots of optional apps and services do require an internet connection, but these are optional parts of the experience. Some examples:

• Skype requires an internet connection, so that it can do video calling and messaging.
• Video streaming services, such as Netflix, Amazon Streaming and Hulu, require an internet connection to stream movies.
• Multiplayer gaming over Xbox Live requires an internet connection to connect the players. Some future games may be multi-player only (eg Titanfall) and so won't work without an internet connection.

this is kind of dependant on what the game does. some games have specific wallpapers for achievements while others use a simple image.

The custom backgrounds was part of the November 2014 update which was around the same time Far Cry 4 launched, so it's understandable they did not have specific backgrounds. It's also a third party game, so I'm not sure that they wanted to put in the extra work in creating images for each achievement.

You can set a custom background to whatever you want, it doesn't have to be an achievement. You can read up on a guide and get a template for custom backgrounds here.

but... to answer your question: no, there is no way to use 360 style achievements on the XB1