Moring wood is a natural part of sleep. During REM sleep the most restful part of sleep that occours usually just before we wake. Males that are functioning normaly get a hard on it is a reflex that occours as part of REM sleep. Nothing abnomal or problematic about it. If you do not have morning wood a few times a week then I would worry. Cause something is wrong.
Generally correct, but unfortunately the world does not work in absolutes.
While absence of morning erections could indicate a problem, that is not always the case. It is an unreliable indicator because anywhere from 10-30% of otherwise normal men have "abnormal" results when subjected to lab testing overnight: i.e., they have weak or absent nocturnal erections.
As long as everything works fine during the day, the absence of morning erections (or more generally, nocturnal erections) is meanigless.