There are indeed limits to what filesystems OS X can handle, as there are with all Operating Systems. Mac OS X only has native read/write access to HFS+ and FAT (although 10.3 still supports HFS, you should be abandoning it because 10.6 dropped write support). OS X does not have write support for NTFS, so you will not be able to write to them. Unfortunately for you, read access to NTFS was only added in 10.3.9, so you wil not be able to see those partitions at all.
FAT32 is another story. This has been natively readable and writable on Macs for a long time, including in 10.3. You should have no problems reading or writing from any FAT32 partition, so it strikes me as odd that you are struggling.
For the purposes of using the disk to transfer files between Macs and Windows machines, you will have to use FAT32 to get native support on both OSes. My suggestion would be to attempt to reformat the drive to use FAT32. There is a caveat here: FAT32 only supports files with sizes less than 4GB. If you are writing large files, FAT32 will not work well for you.
Please note: all of the above relates to native support. There are projects (eg. MacFUSE) that allow the use of other filesystems from a machine running OS X, but I do not advocate them as a solution to this particular problem. They can be...temperamental.
My suggestion for transferring files between Macs and Windows-based machines is to use networking, as OS X uses Samba to implement SMB and so can fileshare to Windows machines. I also advise upgrading to a newer version of OS X, but that's not directly related to this problem.