When you install a game using GOG Galaxy, is there any way for me to also create a desktop icon for my game? The last time I installed one of my games on the library, an icon for the game didn't seem to appear on my desktop. Is there an option that I need to tick or GOG has just yet to implement it?

If you are not using the preview version (1.1), you should find an option to create shortcuts upon a game's installation on the options menu (the little cog/gear-like symbol at the lower-left corner of the client). If you are using the preview version, on the other hand, you'll be given the option just before a game's installation.

The last time it happened, the installation just proceeded right through after the download had done. Does GOG Galaxy update by itself or do I need to do it manually?

When you click a game's "install" button, the game is both downloaded and installed at the same time. If you want to enable shortcuts for the game in question, you have two ways to go about it.

1) Click the options button, which is a cog/gear-like icon at the bottom-left corner of the client. There, you should see an option to enable shortcuts for all games that are installed.
2) Alternatively, you should also find an option to enable preview updates. Click it, close the client, re-open the client and it should be updated. Once updated, try installing a game and you should be given the option to both change a game's installation directory and/or create a desktop shortcut for it.

Oh okay, thanks. I'll try, see if it works.