Why do conditional sentences with "would have had to" have an extra "had" in them?

It gets easier if you concentrate on the past-tense actions:

would have 'gotten wet' ('gotten' implies past-tense)

would have 'had to struggle' ('had to' implies past-tense)

in contrast:

would have 'to struggle' ('to struggle' does not imply past-tense)

The main clause of a "Type Three conditional" has this construction:

past-tense modal (e.g., would) + perfect infinitive (i.e., the infinitive form of the perfect auxiliary have + past participle (e.g., gotten or had))

So it's only natural that your second example has the construction of would have had, because this is exactly the construction of the main clause of a "Type Three conditional".

Could you please clear up the difference between would have and would have had?

If you're comparing would have in your first example with would have had in your second example, then you're comparing two different things, because you should be instead comparing would have gotten in your first example with would have had in your second example, in which case I'm sure you don't see any difference between the two.

Can the second sentence be written without had in its main clause?

If you simply omit had, you get this:

...they would have to struggle tremendously to reach where they are today.

This is NOT the construction of the main clause of a "Type Three conditional", because you don't have the perfect infinitive. So, no, the second sentence can't be written without had in its main clause, unless you're going to make the second example something other than a "Type Three conditional".

The issue here, I think, is a confusion as to the function of have. It can be an auxiliary verb or a lexical verb. In the case of would have gotten wet it is an auxiliary, whereas in would have had to struggle the first instance is auxiliary and the second lexical. In both cases, auxiliary have puts the situation into past time and would marks the counter-factuality of the situation. In the second, had has a meaning similar to must but unlike must it is not a modal auxiliary.

The second sentence cannot be written without had as the have would then be interpreted as the lexical verb and no longer mark a past time situation. The struggling would then be placed in present or future time.