According to 1Cor.11 that is not true. According to all the Scripture on modesty (and there is lots of them), that is not true. God cares a lot more than just the "heart" as you say. However, if the heart is right, then what is on the outside of the heart will also be right.
Perhaps not. You were raised in a generation that has been sadly lacking in culture and etiquette. It is something that the public schools no longer teach. The religious foundation of the public schools is humanism which is atheistic in its nature. The only absolute, they say, is "There are not absolutes." Everything is relative. That is the reason you can say or justify yourself in saying, "It may be right for you to teach your women to wear hats, but it is not right for me to do so." That is humanism. It is ignoring the Scriptural mandate of the Bible and using humanistic rationalism to justify oneself. Here is some evidence to back up my assertions.
Marie Acomb Riley, the wife of W.B. Riley, was the Dean of women at Northwestern College in Minneapolis. During that time she wrote, “Handbook of Christian Etiquette,” published by Moody Press originally in 1945. Robert G. Lee writes the forward. On pages 46 and 47 she writes:
Amy Vanderbilt’s book on etiquette has been a standard for the secular word for many years now. Basic etiquette doesn’t change much. Since I have a couple of books in my library I don’t go every year or so and get the newest updates. Basic etiquette is common sense. Here is what Vanderbilt says in answer to a lady’s question about dressing for church.
“Amy Vanderbilt’s Everyday Etiquette,” by Amy Vanderbilt, 1956, p. 236.
What has been proper etiquette for years, even centuries, just does not change in a few years because the last generation or so have not been taught the social graces. Instead they prefer to either remain ignorant of them, or despise the old fashioned paths that their elders took. It is a mark of rebellion—a combination of both rebellion and the standard of education that is found in public schools of today. I am not sure which. But it remains plain that not only does the Bible teach that women should wear a head covering and men not, in the church; but good etiquette, historically, has always taught the same thing.