Does weapon weight effect how fast you move/run

Yes, based on several types of evidence.

1. I dual-wield exclusively and I know that Fighting Stance, Savage Strike, Dual Savagery etc. stack.
2. The dagger specific perks from the Sneak perk tree stack with the perks from the One-Handed perk tree.

3. The UESP One-Handed Page has a section on dual-wielding which says:

   Allows the use of two perk proficiencies at once, such as Hack and Slash, Bladesman, or Bone Breaker, by mixing weapon types.

   and

   Dual Savagery and Dual Fury can make for almost scary damage output.

As far as I've researched (via wiki's) and tested in game, there is no difference, it is more based on the class of weapon (sword / axe / mace ) than the material structure of the weapon in question.

The materials basically give attributes that make the weapon stronger (coupled with your growing skill progression to use that weapon type).

Speed does not change.

For bows, however, there is a slight difference in draw speed based on the weight. Lighter weapons will shoot more arrows in the same amount of time.