Contributed by anonymous.
Blender files are cross platform - you can play back any Blender file on any operating system on which you can run Blender.
However, a stand alone Blender Player is being worked on. This player will also allow you to make your games 'auto-run'. In this way, your users will not have to install any software but will still be able to enjoy your Blender games.
Update!
Please read https://www.blender.nl/showitem.php?id=187 for more information about NaN's plans for a web plugin and a stand-alone player.
Contributed by B@rt.
Please download ftp://ftp.blender.nl/pub/demo212.zip and study
rigidbodyAndLod.blend.
Contributed by Hideki Saito.
While your game file is open, check "autostart" option from "game" menu, then save your file.
Your game should auto start from next load. You have to open your blender from your shell (either from command line, as in form "./blender foo.blend" and for windows, by either through command prompt or explorer; other-wise, game will start in "editing" mode.
Contributed by B@rt.
No, not yet. (Unless you are willing to program your own multiplayer protocol and server in Python of course).
Contributed by anonymous.
Not at this moment. I wouldn't be surprised if these are planned features though!