For me, the prime characteristic of auxiliary verbs is the traditional function of 'helping' non-finite forms other verbs in the formation of verb phrases showing:
aspect: We are eating, We have eaten.
'code':
...ellipsis: I eat, and so does he,
...tag questions: He eats, doesn't he?
emphasis: We do eat.
interrogative inversion: Do we eat?
'mood': We can/could/may/might/must/shall/should/will/would eat.
negation: We don't eat.
voice: It is eaten.
Within those verb phrases, auxiliary verbs can also show:
person/number: He is eating, they are eating.
tense: He doesn't eat, he didn't eat.
Auxiliary verbs have the NICE properties, but these properties are not an exclusive marker of auxiliary verbs. If they were, every verb in earlier forms of English would be classified as an auxiliary, as would the verb in such idiomatic utterances as How goes it?, Ask not what your country can do for you.
The lexical verbs BE and HAVE [. . .] also conform to the NICE properties, but we will not regard them as auxiliary verbs. The reason is that they can occur on their own in clauses, whereas auxiliaries can't. Aarts, Bas (2011.69, Oxford Modern English Grammar.
It's the NICE properties that mark "be" as an auxiliary verb. In fact, leaving aside one minor construction, “be” qualifies as an auxiliary not just when it is marking progressive aspect or passive voice, but also when it's the only verb in the clause taking a complement with the form of an AdjP, NP etc.
Static "have" can behave as either an auxiliary ("Have we enough money?”) or a lexical verb ("Do we have enough money"). I don’t think there’s much more we can say about that.
Regarding your use of the term 'helping verbs':
Consider a pair of examples like:
[1]
She was painting the house.
[2]
She began painting the house.
"Be" in [1] is an auxiliary verb while "begin" in [2] is not. But this doesn’t mean that there is a difference in grammatical structure. "Was painting" is not a grammatical constituent any more than "began painting" is. "Painting the house" is a non-finite clause functioning as complement of the preceding verb, "was" and "began" respectively. The differences between "be" and "begin" with respect to negation, inversion, etc. (the 'NICE' properties) don’t provide any evidence for saying that there is a difference in grammatical structure, for these properties apply in examples like
[3]
She was ill.
[4]
She hadn’t any money.
No substance can be given to the idea that auxiliaries are "helping verbs": there’s no basis for saying that "was" in [1] is a helping verb while "began" in [2] is not. I would thus strongly recommend dropping the term "helping verb". In this connection compare
[5]
You needn’t come if you don’t want to.
[6] Y
ou don’t need to come if you don’t want to.
"Need" is an auxiliary verb in [5] but not in [6]; there is however, no semantic difference between these, no sense in which "need" is "helping" in [5] but not [6].
"Auxiliary verb" must be defined in grammatical terms, not in semantic terms. Thus auxiliary verbs are verbs with the NICE properties, not verbs that help other verbs: you couldn’t identify them by asking whether or not they help the following verb. They are called auxiliary verbs because they characteristically express meanings similar to those expressed (either in the same language or in other languages) by verb inflection – but this property provides a basis for naming the grammatical class "auxiliary verb" not for identifying individual instances.