NT doesn't like having a FAT32 partition in front of it...
You have the FAT16 system (active primary) partition bit correct... (Basic requirement for dual/multi-boot NT/'9x/etc.)
If you can, using PM4x/5x, convert the FAT32 back to FAT16, and then you should be able to install NT OK to F: (Remember the 2Gig limit for FAT16.) If you still want E: as FAT32 then convert it to FAT32 'after' NT has been installed...
Ideally, put '98 into F as FAT16 or FAT32, and install NT into E.
Usually it's better to have the c: system area as a boot/start-up partition only, with just the NT/'9x (& even DOS) boot/start-up files there; and maybe the i386 Folder (and possibly the '9x Folder too). Then install NT/'9x from there...
C:\i386\winnt /b /x /s:c:\i386 (and press Enter) is the command to install NT from the C: system area. Usually a faster/smoother/less problematic install...
If you're willing to start over...
Make the C: system area around 250Meg, copy the i386 and '98 Folders to it (use a DOS boot-disk with CDROM support or the '98 boot-disk which includes CDROM support...), and then install them from there...
Partition the Extended area of HD-1 for NT and '98 areas. NT can be NTFS later if you wish, '98 can be FAT32 later if you wish. It's not really necessary to have huge partitions for the system files in either case; 2Gig is more than enough, whilst data etc. can be usefully larger if need be...
As W2KStud remarks, your boot.ini points to C: as being the '98 location, not E: . This being so, once you have resolved the FAT32 issue as outlined above, you 'can' leave '98 installed in C: (FAT16) and install NT into either E: or F: - without re-installing '98... This is a popular route to go, but it's far better/wiser to have the C: system for the boot/start-up files (and maybe the i386 etc.) only - as outlined above - and both the actual OS's in separate areas... If you do leave '98 installed in C: then the NT boot/start-up files will co-exist there with the '98 files; and you will still have a dual-boot...
http://www.computing.net/windowsnt/wwwboard/forum/3279.html has put together an interesting aproach to having '98 (FAT32)and NT together - and with '98 before (FAT32)before the NT area...
You could also install NT onto HD-2-and this may allow the FAT32 logical-drive to remain as is.... I'd partition HD-2 into a not too large primary (around 1 or 2Gig max) and install NT there; use the remainder HD-2 for data etc...
You can have NT (system area - the actual OS itself) as NTFS but it's usually a case of over-kill in domestic situations; use NTFS for data etc - if you want to practice 'permissions etc.'; NTFS is hidden to DOS/'9x etc...; FAT32 is hidden to NT/DOS/'95a - without the use of suitable utils (after installation)