Tom,
The 6th ed. rule book specifically says that if the model is holding a weapon that represents something specific, then to refer to the weapon chart on page 61.
"If a model's wargear says it has a power weapon which has no further special rules, look at the model to tell which type of power weapon it has: if it's a sword or dagger, it's a power sword; if it's an axe or halberd, it's a power axe; if it's a blunt weapon like a mace or staff, it's a power maul; if its' a spear or lance, it's a power lance."
So to answer your questions:
1. The showcase Honor Guard that come out of the Marneus Calgar Kit have Power Axes. They get the +1 to Strength, and the AP 2 benefit, while hitting at Init 1, due to it being Unwieldy.
2. "A Relic Blade counts as a power weapon whose hits are resolved at Strength 6." That's exactly how it's worded in the Space Marine codex. Now while the Honor Guard have the option of upgrading to Relic Blades, going by what the BRB says about "Power Weapons", and how they are referred to in the chart provided, I would say that Honor Guard Relic Blades (for those specific models with axes) are resolved at Str 7, AP 2, Init 1. Meanwhile, the Chapter Champion with a Relic Blade, who's modeled with a Power Sword and CCW, would strike at Str 6, AP 3, Init 4.
Hope that helps.