You don't have to convert any existing partitions to FAT16 from FAT32 in order to run Linux. You do generally need to install Linux in a separate partition, at least if you want it to run on its own native file system.
RedHat Linux works best ifyou create a separate partition for it. RedHat Linux can also read from FAT16 and FAT32 file systems, so you can look at files on your Windows ME partition.
If you have the option to put in another hard drive, that is the easiest approach, because you can then format it for Linux without regard to the Windows ME file system. If, however, you do not want to invest in another disk drive, all you have to do is create another partition in the unused part of the FAT16/32 file system.
To do that, first, you want to back up the existing file system (unless you are a gambler), then you want to run scandisk to pack all of the Windows ME files into a contiguous area of space (which tends to improve performance anyway). Then you want to create a new partition, using either FIPS (a free Linux partitioning tool) or a commercial tool, such as PartitioMagic. Actually, at that stage, what you are doing is reducing the space allocated to the existing file system. Then, you use the Linux version of fdisk to create a Linux root partition and a Linux swap partition. Once those are created, you can install Linux. During the installation, you can specify which of these partitions you want to access --- Windows ME partition, Linux Root Partition, and Linux Swap Partition.