"*Any Online only or Multiplayer requests will be denied and deleted."
I would read that as being that as long as the game can be played without an internet connection... it's fine.
The key point is "Online
ONLY".
So for example, if your game has a check to see if a new version was available...
If it did a version check when there was no internet connection and just ran the game anyway... I'm sure that'd be fine.
If it wouldn't run the game because there was no internet connection... I'm sure that probably wouldn't be.
I'm almost certain that using an online check of the version number would be okay, even if that included downloading a patch file so the game can update itself. As long as the gameplay itself didn't need an internet connection. (ie. the game is offline, even if the launcher did an online check).
However, rather than it being an automatic check... you might want to consider making it a [Check for Update...] type button... so that the game never connected to the internet without the player specifically asking it to.
My opinion is mostly based on the fact that lots of Unity games try to "phone home" when they start... Mainly to update Unity's usage stats. But if the game is blocked by the user's firewall... the game still works. Those games make their way to this site too.
People can get a bit twitchy when they notice seemingly "offline" games connecting to the internet. They start wondering what information is being gathered and whether the game includes some sort of malware.
