I tend to agree that those who are overly critical need to look in the mirror... The old kid's response "It takes one to know one" is not true.. However IT does indeed take one to CALL one.
As for removing hats... I hardly ever wear a hot, prefer not to, (Though there are times and places.. )
but I was taught that, for example, when the national anthem plays, you stand, face the flag, remove your hat, holding it in your right hand and placing said hand over the "Heart" (More like the left lung, alas, I know where the heart really is)
When you enter church you remove your hat (Remember when there were clips for holding the hats on the back of pews)
When entering a building you remove your hat.
Actually.. If you read the Military uniform manual.. It more or less describes when you are supposed to remove/wear a hat and save that in civilian life the hat is often optional, and there are a few special cases (Hard hats for example) that you DO wear indoors. The rules are more or less, identical.
And I've had a bit of training in military uniform rules.. Though my uniform did not include a hat since my "Station" was indoors full time. I only stepped out for lunch.