Structure of "As I passed by there looked out from it the face I showed you this afternoon"

First, to make sure that I understand it correctly: could this sentence be rewritten As I passed by, the face which I had showed you this afternoon looked out from it?

Yes, that is a proper understanding of the sentence itself.

Second, is the use of there in this sentence similar to its use in such constructs as there is, or there was?

There can refer to either "something that exists" or "something present in some location".

If I were to reinterpret the original sentence using both meanings of there:

As I passed by, there was a face looking out from it - this is the existential version which is the past tense of "be"

As I passed by, a face was looking out from there - i.e. from that location (which is the window of the brougham)

I would choose the first one as the correct explanation, since in the original sentence the location is already specified as "from it"

Some more examples of both usages of there are found here. Specifically quoting

Existence: I see there are new flowers coming up in your garden

Location: The science textbooks are over there on the floor.

You are right that this use of there is related to there is, there was and similar constructions.

This construction (There + verb + subject) can be used with verbs other than to be—not only with modals combined with to be (There will have to be an investigation), but also (though less commonly) with other verbs, usually where the meaning is one of existence or appearance:

There exist only three examples.

After a while, there appeared a ghost.

Past the window there ran a llama.

First, to make sure that I understand it correctly: could this sentence be rewritten As I passed by, the face which I had showed you this afternoon looked out from it?

No. The speaker saw the face on Bond Street at some time in the past ("one day"), and showed it to the person to whom he is now talking more recently ("this afternoon"), so the word had in your rewritten sentence is wrong. It casts the showing of the face into the further past. If you drop the word had your rephrasing is correct.