Question

Which property affects the strength of a rope?

Answer 1

When the strength of a material is characterized, it is done with respect 
to "force per unit area".  For a given rope material (natural fibers or 
man-made) strength will increase as the cross-section of the rope 
increases.

Regarding the "horizontal or vertical", you are correct; the weight of the 
rope enters the equation as additional load.  However, for most problems, 
you can assume that the weight of the rope is negligible with respect to 
the load.

When thinking about the strength of the rope in real-life situations, the 
rate at which the force is applied is very important.  Consider the 
following experiment: you have two identical ropes with a bucket tied to 
the end of each one.  You add water very slowly to the bucket on the first 
rope, and add it very quickly to the second.  The first rope will 
apperently hold much more weight (appear to be stronger) because of how 
the force was applied.

One factor where the length of the rope will matter is it's ability to 
absorb the shock of a rapidly applied load.  The rope will have some 
amount of elasticity (springiness).  Your 100 foot rope is 100 
times "springier" than your 1 foot rope just becuase of the added length.  
This added length will allow it to withstand a greater shock before 
breaking, but this ability is a function of the "design", not the 
underlying "material strength"

*NOTE* -

Strength does not depend upon length.