I usually try to look at an HQ per approx half a dozen to eight or so stands. For a single nine "fighting stand" battalion plus a couple of supports you might get way with one HQ BUT once you get larger it's a huge chunk of troops left at the mercy of the dice rolls, and distance, and not all wanting to do the same thing.
It might help to think of the battalion as two sections, one under the Battalion commander and one under the senior company commander/XO/whatever, and split them up with appropriate support for whatever mission you're tasking them with on table on the day.
For the big Brit infantry battalions you DEFINITELY need more HQs - you'll barely fit that number of stands within command distance
Rigid is clarified and works as Dr Dave says - the formation the bonus refers to is the entire group of units under that HQ at game start (excepting only recce, AA, suppressed and KO but also unarmed soft skin transport IIRC!).