Most newer BIOS's only require that you have a fat32(or fat16, I suppose)-formated floppy to use. You don't need a dos to boot it, their is a flash utility built into the BIOS.
You boot computer, stick the disk in, and go to the bios entry and have it locate the bios image itself and flash itself.
Other then that there are dos and i386 emulators that you can use to format the disk and make it bootable.
My favorite is
http://bochs.sourceforge.net/ bochs emulator.
Kinda neat actually... the entire computer is emulated, from the cpu to the disk, to the video and computer BIOSes. Completely built out of software. (as opposed to Xen or VMWare that only provide abstraction so that the system thinks it's running in a computer.)
I've used that before to boot up even stripped down versions of Windows 98. Very slow though.
As far as the OS to try out there are bootable versions of MSDOS out their you can play around with, but I prefer FreeDOS. FreeDOS is actively developed and is even used nowadays in modern embedded style systems that don't require a sophisticated system to run them.
If you don't want to muck around with that you can always setup a small partition that boots up Dos and after that you install linux.
Most modern BIOS only need the floppy, or even a cdrom, and can update themselves.
Kinda of a pain even then... since none of my machines have floppies. I have a old spare laying around that I plug into them when I need it.