What I wonder is how I find crash and debug info about this.
/var/log/syslog doesn't mention anything about xmobar, xmonad.errors
is empty. What is going on?
It's going to be in the session log, wherever that is: for `startx` it defaults to the virtual console you ran it on (you can put a redirection such as `exec >~/.xsession-errors 2>&1` near the top of ~/.xinitrc to change it); for display managers it's usually ~/.xsession-errors. Some systemd-based systems may instead use a systemd log accessed via journalctl.
xmonad.errors is *only* used when compiling an xmonad config, not for any other purpose. Because of the specific way this file is used to collect output from compiling an xmonad config, you should not try to use it for anything else.