Redirecting stdout+stderr such that both get written to a file while still outputting to stdout is simple enough:
cmd 2>&1 | tee output_file
>
Order can indeed be preserved. Here's an example which captures the standard output and error, in the order in which they are generated, to a logfile, while displaying only the standard error on any terminal screen you like. Tweak to suit your needs.
1.Open two windows (shells)
2.Create some test files
touch /tmp/foo /tmp/foo1 /tmp/foo2
3.In window1:
mkfifo /tmp/fifo
/tmp/logfile
4.Then, in window2:
(ls -l /tmp/foo /tmp/nofile /tmp/foo1 /tmp/nofile /tmp/nofile; echo successful test; ls /tmp/nofile1111) 2>&1 1>/tmp/fifo | tee /tmp/fifo 1>/dev/pts/1
Where /dev/pts/1 can be whatever terminal display you want. The subshell runs some "ls" and "echo" commands in sequence, some succeed (providing stdout) and some fail (providing stderr) in order to generate a mingled stream of output and error messages, so that you can verify the correct ordering in the log file.