I think the "parallelism" here is potentially misleading. Modern English doesn't normally start a sentence with an adverb in this way. We still use A1, A2, B2 in a limited number of constructions, but the grammar involved is no longer "productive".
As regards sentences starting with "Hardly", I would say these are always dated/formal/literary. Thus, for example, "Hardly had I begun" would normally be phrased today as "I had hardly begun".
We still "tolerate" the form here comes X, but it's no accident that OP chose to switch the verb from arrive to come. Modern grammar doesn't allow constructions like "here arrives the bus", except for certain established usages involving certain verbs (there goes the neighborhood is another one). Here's a chart showing how the form Here stand I, for example, has declined over the past couple of centuries, and here's one for Here stands a man showing that it's the same with both nouns and pronouns.
It seems to me here X comes is an even more "fossilised" form than here comes X. But it's been "modernised" by allowing X to be a pronoun, and we've gotten so used to that form we don't like to put the pronoun at the end any more.
Notice that for the vast majority of verbs which can be modified by here, there, etc., you simply can't put the adverb at the front. You have to adopt the modern style and place it after the verb...
"See that building? I worked there". (grew up, lived, studied, etc.).
Best Answer
In the context of your examples:
The three phrases are somewhat ambiguous. "Here I am" could just as well have been used in the first example in terms of meaning, but it sounds more melodious to have the first two sentences mirror the same construction. "Here am I" wouldn't really fit the first example.
"I am here" could have been used for the second example. But "I am here instead of there" sounds a little trivial (answers "where am I?"), while "here I am instead of there" stresses the "here" to provide better focus on the point of the sentence (a comparison between "here" and "there"). Again, "here am I" doesn't really fit this usage.
Both "I am here" and "here I am" could be used in the third example, but "Here am I" is better. That sequence is like a form of presentation or introduction. It introduces me and then goes on to describe me.
There are a number of other ways the three phrases can be used. For example, "I am here" could relate my location (e.g., uttered while pointing to a map or answering the question, "Where are you?"), or announcing the fact that I have arrived.
"Here I am" could be similarly used, but you probably wouldn't use it while pointing to a map because the map location would be an abstraction while "here I am" would refer to your current actual physical location.
"Here am I" isn't typically used in conversation. The only way I've ever seen it used is to present or introduce yourself as if you where someone else doing the honors. If you're familiar with the old TV program, The Tonight Show, the announcer would introduce the host, Johnny Carson, with, "Here's Johnny!". This is similarly introducing yourself, "Here am I."