The Hat

Jews for centuries have been wearing hats (varying shapes, sizes and colors). It seems that only within the past few centuries the hat has been positioned as headgear of religious significance (based on size, shape and color in more recent years). It is mentioned in a variety of sources that one should wear a hat for davening because one should dress in schul for davening as one would walk in the street. Is this the main reason that the hat has become so significant? We know that in many German schuls it was actually top hats that were worn as in Germany it was the top hat the people wore in the street at that time. 2 specific questions:

When did the hat begin taking on religious significance?
When did jews first begin wearing hats and what information is available on this?
It appears wearing a hat is not Halacha but rather a minhag. Wearing black is mentioned in the Germarah but I'm not sure wearing a hat is.What information is available on this topic?

Steven

When did the hat get introduced as a requirement or at least as a representation of the headgear the Cohain Gadol wore? Clearly people wore hats way back when. I'm interested to know when it took on a significance in terms of how one dresses for davening and thus has become a requirement from the perspective of many.

Steven

If one happens to be davening by oneself, is Minhag Ashkenaz still insistent that one be wearing both a hat and a jacket?

Rallis

Does a käppchen qualify as a hat, al pi minhag Ashkenaz?

In other words, if I am orening in a Chassidic schul, I should wear a kapeljusz? Or cover my head with my tallit in a Sephardi kehillah? What about in a community where the minhag is not to wear a head-cvoering other than the kippah?