Many of these are engineering programs I wrote, myself, in FORTRAN. Windows does not seem to know WTF I am talking about. Upgrade your software yourself, sister.
You could use the newly free VMware Server to create a DOS (or FreeDOS) virtual machine to use with the free VMware Player.
I asked my son, the NT programmer the original question. Answer: no. He said that essentially any DOS program that attempts to interact directly with the hardware will not work. Any other should work, although I wouldn’t bet my life on that. NT (which includes Win 2000, WinXP, and any forthcoming version, but not Win 98 or earlier or that loser version whose name I have forgotten that came out in 2000) does not permit applications to interact directly with hardware and uses something called Hardware Abstraction Layer as the only permitted way.
Delete the program that launches that window (cmd.exe), observe Windows operating normally in spite of its absence, and then see what he says.