What is the origin and exact meaning of 馳せ参じる (はせさんじる)? From what I've heard, it's an old samurai saying, and it means something along the lines of "To go forward in haste". Anyone know?
馳せ参じる is a compound verb made of 馳【は】せる and 参【さん】じる, and means "to flock in haste", "to come at once (to someone higher than the speaker)".
参じる is a variant of 参ずる, which is a humble version of 行く/来る. But this word also sounds archaic, and people usually use 参【まい】る instead today.
馳せ参じる sounds indeed somewhat samurai-sh and/or militaristic, because this word is typically used like "将軍のもとに馳せ参じる", "戦場に馳せ参じる". This verb is too grandiose for everyday conversations, but it's possible to use it in front of your boss or client without sounding funny (as long as you speak very good Japanese).
(to your client) お電話を頂ければ、1時間以内に馳せ参じます。