Will there be mechanical noise when the FJ Cruiser climbs with the rear lock engaged? And will there also be mechanical noise when using the crawl mode for getting unstuck?

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Almost none.

There is a noise when creeping starts, but it stops after the lock is engaged.

Not found.

You need to check if the gear drive shaft's gear ratio is incorrect.

It's normal to hear unusual noises during off-roading, which can come from the body twisting, bumps, scrapes, or tire slippage. However, generally, the differential lock shouldn't make any noise.

Under normal circumstances, a car shouldn't make the kind of mechanical noise you're describing. It's likely that the drivetrain might be slipping.