How to get grep exit code but print all lines?

Use tee and redirect it to stderr

my_command | tee /dev/stderr | grep -q '^Error'

It will save grep exit status and duplicate all the output to stderr which is visible in console. If you need it in stdout you can redirect it there later like this:

( my_command | tee /dev/stderr | grep -q '^Error' ) 2>&1

Note that grep will output nothing, but it will be on tee.

You can do this with AWK:

command 2>&1 | awk '/^Error/{exit_code=1;}/^/ END{ exit !exit_code}'

This will print all the output lines and return 0 if it finds Error and return 1 if it doesn't.