I see a lot of rejected EHLO ... syntactically invalid argument(s). I know it's because of the underscore in the host name, but is there a switch to allow Exim to accept mail from such hosts?

Yes. Add this to your configuration:

helo_allow_chars = _

For more seriously malformed host names, see helo_accept_junk_hosts. See also `FAQ/Policy_controls/Q0732`_.