FAT16 and FAT32 uses attributes like archive, hidden, system, and read-only which Linux does not work with. When you mount these filesystems, you have to specify the umask that will be global throughout the mounted device. You can increase security by specify gid. This will make sure only certain groups can access the FAT partitions. After you mount the FAT partitions you can not change the permissions. Also this goes the same for NTFS but writing to it is experimental. For NTFS partitions, I suggest mounting it as read-only.
In both Linux and DOS/Windows, it is wrong to say folder. The right word is directory.
BTW, there is several questions about this.