How do you boot into a primary parition or the FAT file-system in the first sector of the MBR? That's the error on the USB stick and the error say's that it can't mount the file on linux mint. The specific verion of FAT is either FAT 32 or FAT 16. The interrupt routine and code I've used was something like:

mov ah, 0x02
mov al, 0x01
mov ch, 0
mov cl, 1
mov dh, 0
mov dl, 0x80
int 0x13

Then used dd to write the bootloader/boot sector:

dd if=boot.bin of=/dev/sdb bs=512 count=1

But that command still over wrote the partitions and the formatted FAT-16 or 32 file-system. What can I be missing other then the primary partition offsets and the bootable active flag?

Yeah.
If you really have assembled your above code and copied it into the mbr, than the mbr is of course corrupted. You need at very least the correct primary partitions and the magic code at the end.
Like so: https://en.wikipedia.org/wiki/Master_boot_record