Fallout – Does the damage of a melee weapon, displayed in the pipboy reflect the damage of a regular attack or power attack

No, it doesn't appear to do so. I tested both unarmed and with a pool cue against a group of six ghouls putting one hit on each and did the same damage to each of them.

Other players have tested this, too and can verify that the Gun-Fu perk doesn't affect melee weapons.

This question is more complicated than it appears. Please remember that DPS is largely meaningless in this genre, and not only because mitigation is much spikier than something like an MMO.

If the real question is, "what melee speed is best," well, it depends. For obvious reasons, you want the biggest damage weapon regardless of speed for a sneak attack. For fighting another melee weapon wielder, you almost certainly want a faster weapon than they have, to get as much attacking as possible in between blocking their attacks, and their blocks will mean less per hit.

For everyone else, well, that depends on their armor. The breakpoint for 50% damage reduction is 1:1 damage to DR. For ratios less than that, it ramps sharply up to 50%, and for more than that it slows significantly. (1:20 is ~83% reduction) The upside is, if the average enemy has less DR than your slow weapons, but more or equal than your medium weapons, your slow weapons will do significantly more net damage. If enemy DR is equal to or greater than your slow weapons, it doesn't really matter what you use.

The point is, there's no hard or fast rule to determine melee dps. I wish I could just give you a coefficient for each speed to divide damage by, but it just doesn't work that way.