•する  to do
させる to make someone do
させられる to be forced to do
•勉強する to study
勉強させる to make someone study
勉強させられる to be made to study by someone
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First of all, だまらせる is the causative form of だまる. And I think this sentence is Causative-Potential. And こいつが黙らせられる is also Causative-Potential in this sentence.

I think が黙らせられる becomes Causative-Potential or Causative-Passive according to the context. For example, 私に、彼が黙らせられるかな? ("Can I make him be silent?") is Causative-Potential. 私に彼が黙らせられる ("He is made silent by me") is Causative-Passive.

が and を are commonly used in potential form, but が is more appreciated than を. For example,　私は本が読める、私は本を読める
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If intransitive verbs don't have it's transitive counterparts, use the causative form to express something similar

First, 作動する is almost always intransitive, whereas English run is both transitive and intransitive. In English, you can safely say both "the system runs" and "run the system".

In Japanese, you can say 仕掛けが作動する (intransitive) but usually not 仕掛けを作動する (transitive)

So 仕掛けを作動させる literally means "(I) make the device run/function", not "(I) run the device".
(SO THE DEVICE RUNS BY ITSELF, BUT THROUGH ME (BECAUSE I MADE IT RUN))
The subject of the verb (作動させる) is the speaker, which is omitted in this sentence, as usual.

私 は/が 仕掛けを作動させる    (を is interchangeable with に because 作動する is intransitive)