Your first question is not really Perl related, it is an issue with the shell (cmd.exe) you are using. For example, Bash has an array of pipe return codes, however this is quite difficult in shells like csh and ksh. I don't think you can do it in cmd.exe.
However, there is an alternative using Perl. Run your build.exe in a pipe:

open(my $fh, "C:\\IA71_Enterprise\\build.exe $ProjFile 2>&1 |") or die
+ "$!"
[download]

and do the tee yourself in Perl, you can get the return code from $?.

The tee utility in DOS can't redirect the STDERR output to a file
You answered this question yourself in your system command. Redirect stderr (Windows file handle 2) to stdout (Windows file handle 1) before the tee using 2>&1.

By the way, this might be being pedantic, but I doubt you are actually running a DOS program. PC or MS-DOS was 16-bit, you are probably running a 32-bit console program - at least, I hope you are!