No. ASM is just a slightly easier mnemonic version of the raw machine language. All software basically compiles down to this form.
The OS functions are provided here. The "game" is just an application in the OS (kinda). The BIOS provides whatever the OS needs to hook into to use the hardware. Here's a high-level overview of how this works http://duartes.org/gustavo/blog/post/how-computers-boot-up/
OS's don't need to be very complicated, just a few bytes in some cases depending on what you want to do. For example, you can just use the standard VGA modes (here, he's using 13h) to get something to display, and just write directly to the video memory. The BIOS helps handle some of the interface work to the various standard components.
This is the real old-school way of doing things and the code is really well done and commented. If you know x86 asm basics, you can read through most of the system-level stuff without too much trouble.