2

Is a creature with double strike effected by 'creatures lose first strike and cannot be given first strike', for instance if an opponent has Archetype of Courage?

6

No. First Strike and Double Strike are entirely separate abilities.

First Strike and Double Strike are different abilities with overlap in their effects. Effects that care about First Strike ignore Double Strike, and vis versa.

From the rulings on Urborg (which has the ability "{T}: Target creature loses first strike or swampwalk until end of turn."):

Urborg’s second ability affects only the first strike ability, not the part of the double strike ability that acts like first strike.

3

No, every keyword ability, even those that function similar ways is a different ability, and effects that modify one would not modify the others.

If that were possible, Archetype of Endurance would allow opponents, but not the controller, to target creatures with shroud, since the hexproof half would be removed.

This is also the case for lifelink and lifelink like abilities like Spirit Link. While they behave in similar ways, they are not the same and changes to one do not change the other.

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