According to Manhattan GMAT, the following are incorrect.

I'm happy to help with these.

"Following the gold rush, the mining town collapsed." (The correct version is "After the gold rush, the mining town collapsed.")

Technically, the word "following" is a participle, and therefore it should "touch" the noun that is doing the following. Thus,
"Following the dog act, the pony act was a disappointment."
The pony act is the actually think that "followed", that did the "following."
In the sentence above --- yes, that sentence would pass as perfectly correct in colloquial language but technically, the "mining town" did not perform the action of "following" anything.

"We adopted new policies with the aim to reduce theft." (The correct versions are "We adopted new policies aimed at reducing theft" and "We adopted procedures with the aim of reducing theft.")

To "aim at" is correct.
To speak of an "aim of doing X" is correct.

The construction "aim" + "to" is technically not correct. Again, you will hear this in colloquial speech, but that's no guide --- colloquial speech is littered with errors. I don't know that I can "explain" anything besides making the rule clear.

"When compared to horses, zebras are viscous." (The correct versions are "In comparison with (or to) horses, zebras are viscous", "A zebra can be compared to a horse in many ways", and "Compared with a horse, however, a zebra is very hard to tame.")

First of all, I believe the word you want is
vicious -- wild, dangerous, not easily tamed,
and not
viscous --- (of a fluid) thick, syrupy, not easily flowing

The first one makes the famous "missing verb" mistake. The GMAT brutally punishes the following construction
[conjunction][participial phrase]

You see, a conjunction (like "when") must be followed by a full bonafide noun + verb clause, NOT simply a participial phrase. See these blog articles:
https://magoosh.com/gmat/2012/participle ... -the-gmat/
https://magoosh.com/gmat/2012/gmat-gramm ... b-mistake/

Thus, the construction would have to be "When they are compared to horses, etc.", but that's now awkwardly long and indirect.

Does all this make sense? Please let me know if you have any further questions.

Mike