Learn English – “Starting sentences with Adjectives “

The means "one X we've talked about or seen before."

It means exactly one specific known/seen X - if there is more than one X, you can't use the unless you qualify the noun with adjectives so we know which one. The X can refer to a plural noun that is an indivisible group logically functioning as one "unit", i.e. the dozen eggs.

It also can be used with nouns that fall into a logical category of "one X that everyone should have seen or heard of before" or "only one possible X in the world."

Introducing a singular noun into a conversation with the indicates the speaker expects the listener to know which one on his/her own.

It's difficult to give general rules on when to use the versus a because the points above heavily depend on variable conditions outside of the conversation.

Walking around Prague I saw the house that was painted red.

Is there a well-known red painted house previously known or seen both the speaker and listener? We then use the to indicate we are referring to it. If a was used, it means a red house was seen, but it wasn't anything special, and it wasn't a specific one known or seen before by the speaker/listener.

English, as you know, is full of exceptions. One of those is generally you do not use the in front of proper nouns that are the names of people.

Technically, you can do this, but you will be making the person sound "legendary" - as though he/she has done something so grand or unique that "one X that everyone should have seen or heard of before" or "only one possible X in the world" applies. So, in that case the Queen Elisabeth could work.

Yes, sort of. A determiner, which includes articles, has to be there. Other classes include, possessive pronouns and nouns (e.g. my, your, John's) and demonstratives (e.g. this, that). Something, often an article, has to grant that shirt a sort of relative presence in the universe. Thus the red shirt, a red shirt, John's red shirt, that red shirt -- they all work. What does not work is red shirt with no word summarizing how it fits in the universe.

I understand that certain other languages do not require determiners. Or rather, context or some grammatical structure besides determiners accomplishes the same purpose. It is difficult to imagine a language which had no way of differentiating that red shirt from Tom's red shirt from any red shirt.

Inspired by Satnam's answer, let's look at plural nouns:

Do you see boys in red shirts?

Note that neither of the plural nouns has an obvious determiner. How can that be? It would appear English sometimes uses the context of plurality to apply an invisible or assumed determiner. The words any, the, and some would be good candidates. These vary in subtle ways:

1. Do you see any boys in any red shirts?
   The possibility exists that there are no red shirts, and perhaps not even any boys.
2. Do you see the boys in the red shirts?
   Both boys and red shirts exist.
3. Do you see some boys in some red shirts?
   This case has strong implications that there may be girls or adults, and that some of the boys' shirts may not be red. Neither of the first two rule out those possibilities though, nor does this case require them.

Any vagueness of meaning could have been avoided by choosing actual determiners instead of leaving it to the listener or reader to decide. But if it doesn't matter to the author/speaker, so be it.