Is it possible for malware to be written in the BIOS section and loaded on every fresh Windows Install? If it is possible, can this affect Linux (Ubuntu and others) operating systems as well?
Yes, the BIOS is software. If it is writeable/flashable, it is possible for the BIOS to be overwritten or modified such that it performs malicious actions. The BIOS boots before the operating system and bootloader, so it is entirely possible for BIOS malware to impact and infect on each boot or on fresh installs.
This is not a windows specific problem. The BIOS provides the lowest level hardware interaction/abstraction and the OS will depend upon this code, so it may be possible to write generic or OS specific malware.
The MITRE Copernicus tool can be used to do checks for malware in BIOS.
Yes it is possible although newer bios has code signature protection which make this attack unprobable. The BIOS is tied to the computer architecture in this case x86/x64 and it's end goal is to load the OS. So no matter the OS even no OS you can flash (write) the BIOS.
However, the BIOS is not a malware delivery channel. For any useful work (accessing virtual memory, the file system, the network etc..) you'll need to run the malware code after the boot loader. The first stage boot loader has the same limitation and has only 640kb of space so you couldn't put much malware there. The second stage boot loader would be the first location where you could execute useful malware code after the filesystem has been setup. It is more practical to develop a device driver to deliver low level malware and leave the boot process untouched.