[Author -- hang on, interviewer here]
Yes it supports FAT32.
No, DOS never supported file sharing on its own. It was always an optional extra. That's why Novell invented Netware Lite, then Personal Netware, then bundled it with Novell DOS 7. It's why SAGE sold MainLAN, and why in the USA, LANtastic and the $25 Network made livings.
Yes, one MS client could share too, but it took a lot of conventional memory so it wasn't much use, and that's the main reason it was removed and moved into Windows for Workgroups.
But it was never a standard function, and it was not connected with FAT16, which appeared with MS-DOS 3.0 for the PC-XT and PC-AT which had hard disks.
https://en.wikipedia.org/wiki/File_Allocation_Table#FAT16
You may be confusing the DOS redirector, which appeared in MS-DOS 3.1 and was mainly used by MSCDEX for CD drive support.
https://en.wikipedia.org/wiki/MSCDEX
FreeDOS can run the MS client, and the IBM client, and both Novell P2P clients, AFAIK. However none of them natively talk TCP/IP so they are not much use today.
You can add TCP/IP to them but the result is a lot of memory usage -- I have tried it. Performance varies widely, too, as the OS/2 Museum measured:
https://www.os2museum.com/wp/dos-smb-client-performance/