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Speed: When making a full-attack action, the wielder of a speed weapon may make one extra attack with it.
Vital strike says that you are able to make one attack with the attack action. The weapon property and the feat are somewhat at odds with each other. I think the feat would limit the number of attacks to one.
Yeah, you can't get the extra attack from the Speed effect when using Vital Strike because Vital Strike requires a Standard Action, and isn't a full-attack action.
However, Power Attack doesn't require an action to use, it just changes how your attacks work, so it works with either.
As mentioned Power Attack works with both Vital Strike or a Weapon of Speed ability (ie Haste), but Vital Strike and Weapon of Speed do not work together.
Vital Strike is a specific full-round action in which you forfeit your extra attacks for double damage, so unless you get two rounds per round which might happen with wierd timey whimey stuff, you can only get one attack. Power attack should be able to apply to either though.
That is wrong.
Vital Strike requires you to use the "attack action". The attack action is a special kind of Standard Action in which you make a single attack.
The feat itself is neither a special kind of standard nor full-round action, and can therefore be combined with anything that affects normal "attack actions" aka single attacks.
It can not be combined with a "full-attack action" which is a special form of full-round action, since that's not an "attack action".
It can also not be combined with Charge or Spring Attack, since those are actions on their own that allow you to make a single attack, but are not an attack action.