[philman_36]

Is it customary to allow a foreign government access to a US fighting ship for exam of the mechanical systems?
AFAIK, no. A pilot is the only instance I know of where a foreigner is allowed aboard a ship underway that I know about.
In the case of an accident...I can't say. No experience there.

A ship usually goes through battle drills/training before deployment. Such problems are usually found then.
A ship is given a 'Battle E' if they do well.
https://www.flickr.com/photos/navalsurfaceforces/16394672011/in/album-72157633023873179/
Zoom in and you can see their Battle E. For ships, it's a very big thing to be awarded one.

[GreyHoundSailor]

A Destroyer being awarded a ‘Battle E’ (the white one) indicates they are the best DDG in the Squadron, not just that they ‘did well’.
There’s a lot that goes into it, and they have to be awarded at least four of the Command Excellence E’s (the black, red, blue, yellow, purple ones).
I served on four ships and have four Battle E’s. We never hit anybody either, although another Navy ship did hit us while we were moored pier side in the US. Also once watched an Italian Destroyer disappear in the shadow zone of our CVN - that was exciting.