Since you've just regurgitated point after point without a breath, I'll respond in kind:
- Calling "Sarah" "Sarah Jane" has no impact upon alphabetising.
- Of course we drop "Madame" (per T:HONOR).
- Yes we drop the Brig's name (again, per T:HONOR, as Shambala108 explained).
- Yes K9 is listed as a television companion, because the current structure of these templates stresses the media of origin.
- Having "Jack, Jackie" isn't confusing.
- It's not the job of this navigation template to indicate familial relationships
- Yes, Christina. In the context of list that's labelled "Original to television" in a navbox labelled "Companions of the Tenth Doctor", it won't be confusing at all. There's only one Christina.
- K9 I, K9 II (television) K9 III (prose)
- The only person whose name I'm thinking of hiding is the Brigadier's. And that's an exception already covered by rule. You're being hyperbolic when you say "hiding parts of some people's names but not everyone is a big confusing deal". The rule is simply to go with the one name by which they are most commonly known. That's not confusing because it's consistent.