Let’s say you have some command and you are interested in its result code and output in the SAME TIME. Problem is that when you use $() construct to get output of some command, then you do not have any way to get its result code.

Well, in that case, why not use pipes and read command? See:

some-command | read command_output

echo "Result code of command is: $?"
echo "Output of command is: $command_output"

read reads input from the pipe and stores it in command_output variable. Simple, works.