Viking sword, Rapier, Sabre, Cutlass and Katana, which is harder and lighter (show the weight of sword if can)

Answer 1

Rapiers as a class are the lightest of the swords, though each sword individually varies.
Small swords, which are a sort of 'descendant' of the rapier were lighter still, being optimized for dress wear and a somewhat ritualized formal combat.

Sabers, depending on what type are also very light, around the same weight as many rapiers. Cutlasses were generally heavier, and both Katanas and 'viking swords' were in about the same weight class. Historical examples show that on average the swords weighed between three and four pounds.

Though optimized for very different purposes, each of the types of swords you mentioned were excellent for what they did.

An excellent essay on medieval period swords can be found here: http://www.thearma.org/essays/weights.htm with references aplenty.

Answer 2

rapier < sabre < cutlass < katana

I don't know exactly what the vikings used--maybe a two handed sword similar to a katana.