For now, Pentagon and DHS won't recommend that Trump invoke the Insurrection Act

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12. No. It's a power delegated by statute to the president.

10 USC 253: Whenever the President considers that unlawful obstructions, combinations, or assemblages, or rebellion against the authority of the United States, make it impracticable to enforce the laws of the United States in any State by the ordinary course of judicial proceedings, he may call into Federal service such of the militia of any State, and use such of the armed forces, as he considers necessary to enforce those laws or to suppress the rebellion.

While invoking the Insurrection Act to allow the use of the military to perform civilian law enforcement activity is not a common occurrence, it is something that has been done multiple times in my lifetime when the use of military to supplement law enforcement is deemed necessary to suppress riots and/or looting. For example, both JFK and LBJ invoked it several times. The most recent example is GWBush invoking it to allow military to assist in suppressing the LA riots following the Rodney King beating.

It would be a stretch for it to be invoked in the absence of actual rioting/looting, but it wouldn't be unprecedented either.