How do I extract the ROM file from this DOS exe BIOS updater?

Actually that was trivial, simply run it under wine:

$ wine MS-A10.exe

The output will be found in .wine directory:

$ find $HOME/.wine | grep MS-A10
/home/mathieu/.wine/drive_c/users/mathieu/Temp/MS-A10
/home/mathieu/.wine/drive_c/users/mathieu/Temp/MS-A10/AFUWIN.EXE
/home/mathieu/.wine/drive_c/users/mathieu/Temp/MS-A10/AMIFLDRV.VXD
/home/mathieu/.wine/drive_c/users/mathieu/Temp/MS-A10/AMIFLDRV64.SYS
/home/mathieu/.wine/drive_c/users/mathieu/Temp/MS-A10/AMIFLDRV32.SYS
/home/mathieu/.wine/drive_c/users/mathieu/Temp/MS-A10/RosaBUU.exe
/home/mathieu/.wine/drive_c/users/mathieu/Temp/MS-A10/Pmxdll.dll
/home/mathieu/.wine/drive_c/users/mathieu/Temp/MS-A10/Idrvdll.dll
/home/mathieu/.wine/drive_c/users/mathieu/Temp/MS-A10/FWUpdLcl.exe
/home/mathieu/.wine/drive_c/users/mathieu/Temp/MS-A10/settings.ini
/home/mathieu/.wine/drive_c/users/mathieu/Temp/MS-A10/error.log
/home/mathieu/.wine/drive_c/users/mathieu/Temp/MS-A10/1176upd.BIN
/home/mathieu/.wine/drive_c/users/mathieu/Temp/MS-A10/AFUWINx64.EXE
/home/mathieu/.wine/drive_c/users/mathieu/Temp/MS-A10/RosaBUU.log
/home/mathieu/.wine/drive_c/users/mathieu/Temp/MS-A10/MS-A10.BIN
/home/mathieu/.wine/drive_c/users/mathieu/Temp/MS-A10/FWUpdLcl64.exe

with:

$ file MS-A10.BIN
MS-A10.BIN: Intel serial flash for PCH ROM

I guess this has little interest for anyone trying to do this BIOS update, you should be aware that using a Win7/32bits repair disk is not an option (I followed instructions from here). Indeed the DLL needed at runtime requires that oledlg.dll be found:

$ objdump -x RosaBUU.exe  | grep DLL
 vma:            Hint    Time      Forward  DLL       First
    DLL Name: dbghelp.dll
    DLL Name: KERNEL32.dll
    DLL Name: USER32.dll
    DLL Name: GDI32.dll
    DLL Name: COMDLG32.dll
    DLL Name: WINSPOOL.DRV
    DLL Name: ADVAPI32.dll
    DLL Name: SHELL32.dll
    DLL Name: COMCTL32.dll
    DLL Name: SHLWAPI.dll
    DLL Name: oledlg.dll
    DLL Name: ole32.dll
    DLL Name: OLEAUT32.dll

Apparently this DLL is not present from a Win7/32bits repair disk.

There is a command-line switch to extract the ROM (may require Windows):

MS-A10.exe /writeromfile