Empire AB p. 30: "Detachments are trained to support their Regimental Units and protect them in battle. If an enemy unit declares a charge against a Regimental Unit, and the Regimental Unit does not flee as a charge reaction, then each of its unengaged Detachments within 3" can make either a Support Fire or a Counter Charge action."
And that is it - there are no rules allowing a Support Charge.
2. What you said about Support Fire is what I said about Support Fire.