1) Is it normal for games to install without a password being asked for?

2) After the latest kernel update, I get the message on startup saying I have 0 bytes left in boot. What is that about?

1. If the game is installed for only active user (like GOG games) yes it's normal. The game will be installed in user home directory

2. Check your free space

For question 2, linux maintains a separate partition called boot where kernels get installed, that partition can run out of space even if your hard drive otherwise has plenty of space left.

This post about Ubuntu describes a similar issue.