Tension vs Tensile Strength A Brief Explanation of the Issue
A common problem type for vector calculus students or introductory physics/engineering students is tension problems. For instance, if a mass of $10$ kg is suspended from two taut cables of length $3$ m and $5$ m at angles of $52^\circ$ and $40^\circ$, respectively, then the corresponding tensions in the cables are $75.12$ N and $60.37$ N.
To make this concrete to students (in terms of application), one might talk about the strength of the cable material, e.g., would this arrangement break either of the cables. Looking up material strengths, these are most often measured in kPa. So from my understanding, the 'failure' in this situation  is a measure of resistance to pressure in terms of tensile strength.
The Question
How would one determine if catastrophic failure of the cables would occur? Naturally, this criterion is expressed as a pressure, but what is the area I would be using? Would this be the area of a slice of the cable attaching the cable to the object and to where the cable is 'grounded'? Would it be the area of the cross section where the cable is thinnest? Is it an average size of a cross section of the cable? While I assume the tension is constant in the cable as a whole, the pressure from the object is not constant throughout the cable, correct? Or given that the cables are at an angle, would some analysis of the forces from a cable's tendency to rotate until it is vertical due to the weight weight be more relevant, i.e., some type of twisting force? If anyone could clarify the situation in this case, or more generally, this would be wonderful.
[Note: I am a mathematician, with only an undergraduate background in physics, if this helps with the type of explanation that would be most understandable to me or my students.]
Considerations
I teach college students, typically freshmen/sophomore engineering students who have taken their calculus sequence up to multivariable/vector calculus but have had only an introductory physics course—if any at all.
 A: 
Looking up material strengths, these are
most often measured in kPa. So from my understanding the 'failure' in
this situation is a measure of resistance to pressure in terms of
tensile strength.

The term usually used is yield strength, which is the maximum tensile stress before a specified amount of plastic deformation is produced, usually taken as 0.2 percent of the unstressed length. Tensile stress is the more applicable term than "pressure", though the units are the same.

How would one determine if catastrophic failure of the cables would
occur?......(etc.)

There are really too many questions here to answer point by point. What's more, cables are a special application of materials, as opposed to general applications such as columns and beams,  and there are probably specific standards on how to test the tensile strength of cables. It does appear that the cable diameter is used to qualify the tensile strength, as in the following example link. https://componentsupplycompany.com/Stainless-steel-wire-tensile-strength.php
Just to put things in perspective, the figure below is a generic stress strain curve which gives you terminology related to the various levels of strength of materials, if you will.
Proportional limit: Basically this is the maximum stress where the stress-strain curve is linear and follows Hooke's law. When the applied stress is removed the specimen returns to its original length (no permanent deformation).
Yield strength: This is what I referred to in the beginning. It is the maximum tensile stress before a specified amount of plastic (permanent) deformation occurs.
Ultimate strength: This is the point where the specimen can no longer withstand any further increase in applied stress, the curve turns downward and failure (fracture) is imminent.
Fracture: Here the specimen fails.
If you are going to teach this stuff, you should probably apply it to general principles and applications of mechanics of materials, rather than the specific application of cables. There may be a separate science associated with cables, but I don't know.
Hope this helps.

A: One approach
Teach the tension and tensile strength concepts over several lesson plans.

*

*Lecture on tensile strength of different materials, and how that tensile strength is measured in the lab


*Schedule a hands-on lab where students get to actually measure the tensile strength of a few different materials.  This would probably require thin wires of different materials, a micrometer, some weights, etc., with the usual amount of instruction on how to use the equipment


*Lecture on static equilibrium, with example problems illustrating how a given weight can put varying amounts of tension on suspension wires, depending on the angles involved


*Schedule a hands-on lab involving the various angles encountered in step 3 (above), with spring scales, load cells, or some other force measuring device, that will allow students to actually measure the effect
The combination of lecture first, immediately followed by hands-on experience, should give students a good idea of what is happening.
Now, to directly answer the questions.
Catastrophic failure occurs when the force on a cable exceeds its tensile strength.  This failure will occur at the minimum cross-sectional area of a cable, if the cable's cross-sectional area varies (and it will vary, even if on a microscopic level).  Also note that for the usual situation, the weight of the cable is not significant, so you can assume constant tension in the cable.  A review of intro physics literature on this subject should answer your questions in quite a bit more detail.
A: If they are engineering students, it is interesting also to present the type of cable that are normally used for lifting loads: wire rope.
Except for small loads, it is difficult to have a steel cable that is at the same type strong and flexible. The solution is a bunch of wires hold together.
In principle, the tensile strengh of the rope is the sum of the strenght of all of the individual wires, but it is better to follow the manufacturer specification, that takes in account the friction  and the fact that some of them can be damaged during its lifetime.
They can fail suddenly if the load is above the specification, but also if they are badly damaged and people don't inspect to discard them when necessary.
