The two uses of "be"

I am reading a book about English verbs. The first chapter gives some basic definitions, and said that "be" can be used as a lexical verb and as an auxilliary verb. For example: "I am tall" or "I am running"; or "He is happy" or "He is reading".

My question has several parts:

1) Why can the "be" verb be used in these different ways?

2) Why does English require this "helping" verb for the progressive tenses. When you think about it, what is it really "helping?" It seems only useful in the sense of distinguishing the simple tenses from the progressives. Why not say "I now read" or "I now reading" and use "now" or some other word as the "helper?"

3) When you say "I am [word]," you are saying the subject has this quality, is in this state of being, or is the word. For example "I am tall," means you have this characteristic, "I am sad," means you are in this state of being, and "I am a fireman" means that you and fireman are the same. So, if you say "I am reading," it seems that you are somehow saying that you and the activity of reading (the gerund sense of the word) are the same. There seems to be a kind of relationship between you and "reading." Is this what's really going on but is just invisible to must of us? Is there some history to the use of "be" that addresses this (or is this more related to the development of "ing" in English?)

You mustn't believe all you read, especially about grammar. Why do people sometimes say that "be" can be either an auxiliary or a "lexical" verb? I don't know what "lexical" is supposed to mean here, but that's what they say. But why?

Well, here's my theory. "Auxiliary" in this context means "helping verb", and sometimes there is a following verb that you can imagine that "be" helps, like the present participle of a verb in the progressive aspect. But sometimes there is no following verb for the "be" to help, in which case it can't be an auxiliary and so it must be a "lexical" verb itself (since as we all know, every complete declarative sentence must have a verb).

Does that make sense? Alas, no, because it's not based on any facts of language, but instead, solely on terminology. Maybe we call the "be" that goes with an accompanying verb an "auxiliary verb", but that doesn't prove anything about that "be". Maybe "auxiliary" is an appropriate term, but maybe it isn't.

I could declare that all nouns beginning with the letter "k" are a special part of speech called "kymniad". Would you believe me?

According to the only syntactic tests I know of for what is an auxiliary, the "be" that accompanies another verb works just like the "be" that doesn't. For instance, auxiliaries are inverted in order with a sentence subject in questions, unlike ordinary non-auxiliary verbs:

You are eating. -- Are you eating?
You were seen. -- Were you seen?

Ordinary verbs don't work this way in contemporary English:

You eat halibut. -- *Eat you halibut?

So, we might expect the "lexical" verb "be' to work like the ordinary verb "eat", right? But it doesn't -- not at all.

Are you a halibut? (not *Do you be a halibut?)
Are you in the kitchen? (not *Do you be in the kitchen?)
Are you purple? (not *Do you be purple?)

So where is the language evidence that there is a grammatical difference between "auxiliary be" and "lexical be"? Nowhere.