Of course - generally the expectation is that control surfaces will not move on their own accord, unless specifically asked to; either directly by the pilot by moving the yoke/stick, trim wheel, rudder etc, or indirectly by setting the autopilot, which in turn moves those controls.

If they move on their own accord - usually via autopilot - it's because they are trying to maintain a reference point the pilot asked them to.

When they move on their own accord, outside of any pilot inputs or settings, without the pilot knowing / understanding why, that's where the issues start happening.

Air France AF447 entering Alternate Law, Lion Air MCAS activating etc.

The initial course of action is always to return to stable flight, disabling whatever systems you perceive to be changing the control surfaces. When you just don't know which system is causing the issue, then you're just fighting the end result (eg nose down), not the underlying system issue (eg MCAS preventing stall by enforcing trim).

To find the cause of the end result (eg nose down), you work backwards to see what happened - because, chances are, you as a pilot made an erroneous setting / configuration somewhere to cause an undesirable aircraft state, not the system itself, as you are taught the system works under your command, plus no system acts to down an aircraft - they are all there to prevent an aircraft from getting into an undesirable state. You go back to the last known working configuration, and work from there to diagnose.

Long winded answer, but yes, you do check control surfaces, but they only show an undesirable aircraft state which can be corrected (eg pull back on the stick), but it doesn't solve for the what and why, and thus preventing getting back into that undesirable state. Trim is also one of those ones right at the bottom to check, as it's for fine-tuning an aircrafts pitch - where an elevator can move an aircraft up or down tens of degrees in a seconds, trim usually only moves the pitch tenths of degrees.

This is also where SOP's and training come in too - some airlines would have "set landing trim" on the final leg of the approach, others might have it on the base leg. Some companies require stall training in the sim once a year, other airlines don't even have sims. Some airlines are okay with reverse thrust on landing, others want to avoid wear and tear on the engine and want you to slam the brakes. The aircraft manufacturer would have their "recommended" training, and modify it in accordance to company practices / requirements, legal / regulatory frameworks etc.

From what I understand, and this is just from reading the news and forums, there was no knowledge or training about the MCAS system. The closest thing is a runaway trim, and again this probably differs from airline to airline.

All aircraft, pilots have a checklist of things - takeoff, landing, recovering from various undesirable aircraft states etc. Occasionally you do hit an edge case where a lack of knowledge and training collide with a background-running aircraft system acting out of the ordinary.

Again - I'm just a private pilot so do take the above with a grain of salt, Airline Transport Pilots have far more training and experience, and can probably provide a far better insight to the behaviour of larger jets, difference between companies, SOP's etc.