If you are sharing the drive on a network the filesystem doesn't matter, as long as the host computer can handle it.Most NAS appliances use filesystems that Windows wouldn't recognize.
Fat32 would be the very worst choice.
Assuming, of course, that OP isn't trying to connect the hard drive directly to a windows box rather than through the ubuntu one in which case the ext4 would be the worse choice.
Best option if you don't need to save what is in the Lost+found folder is to simply reformat the drive. If it is only going to be physically connected to the Ubuntu machine go with ext4 but if it may be moved and physically connected to a windows machine then FAT32 is the better option.
"All I want to do is put that drive available to all computers on my network."
It seems a fair assumption to me that, with a media server, a networked drive is required rather than a box than can be unplugged from the Linux box and plugged into the Windows ones. That wouldn't make a lot of sense when you have a network available.
Rereading the OP, I'm a little unsure of the current situation. You say "it won't show up in computer", and then "when I go to the directory mount point", which means you can see it. Forget "lost+found"; that is a directory that appears with any drive formatted with the standard Linux filesystems (it's for recovering data when a disk is corrupted) and normal users should not be able to access it.
It sounds to me as if you can actually see the drive (appearing as a mounted directory rather than a separate drive is how Unix works). You want to share that directory to the network so that the other computers can map it as a drive. For this you need to use Samba. Have a look at this link for details of installing Samba and using it to share files to other networked computers: https://help.ubuntu.com/10.04/serverguide/samba-fileser...
You say "it won't show up in computer", and then "when I go to the directory mount point", which means you can see it.
Which is what had me suspecting that he was moving it around.
In his image it shows Lost+Found occupying nearly the entire drive. Maybe I'm wrong but I've not seen that before. If there is something that needs saving he needs to open it. If not a reformatting is a way to go in which case he can follow your links.
In his image it shows Lost+Found occupying nearly the entire drive. Maybe I'm wrong but I've not seen that before.
I hadn't actually looked at the screenshots, but that certainly makes things clearer. It doesn't show the lost+found folder occupying nearly the whole drive.
It's showing the drive as being mounted on /home, so yes it is visible. The properties of the lost+found folder shows 434.9GB free space (i.e. the disk is essentially empty). That is not the space occupied by the folder, which will be 0GB unless fsck has been done on a corrupt disk, but the free space on that disk.
/home may not be the best place to mount the disk, as will hide any existing /home directories, but it's certainly a possible mount point if that's what's required. All the OP needs to do to share the drive to the other network computers is to install and configure Samba. He may also wish to choose a different mount point; other than that the disk is partitioned and formatted and everything is fine. I'd recommend that he read the two articles I linked to earlier.