Boot process of atari (tech info)
The atari ST has its bootstrap, BIOS, and GUI in ROM (well any atari ST after ~1986)
No floppy disk is needed to boot to the desktop, though if there is no hard drive, a blank floppy in the drive will speed up the boot process considerably.
Linux is not available for the basic ST (something does exist but I nothing "mainstream" i think).
If you write a program that runs under 68K then compile it and place it on the floppy in a folder called "AUTO". When the system boots it will run this program first. Other than that, you can put code in the bootsector of the floppy, and make the checksum of the sector $1234. This will boot earlier in the startup sequence than the auto folder one.