Learn English – the difference between “outnumber” and “outnumbered by”

Sentence #1

I remember being taken to the zoo.

This is correct and sounds very normal to native ears. Being taken to the zoo functions as a noun phrase denoting the act in which you were taken to the zoo. Being is a gerund here. The passive construction here omits the subject; the implied subject is me (which a person would only say in order to create unusually strong emphasis).

It might help to look at some comparable sentences where the subject of being taken is explicit:

I remember John being taken to the zoo.

I remember him being taken to the zoo.

Notice that the subject of being taken is in the objective case. That's because it's also the object of remember, the main verb of the sentence.

Sorry, I have to tell you this

There's also an older school of thought that says the previous two sentences are incorrect and should instead be:

I remember John's being taken to the zoo.

I remember his being taken to the zoo.

In this parsing, the object of remember is being taken to the zoo. The subject of the gerund in this construction takes the possessive case (strangely enough).

Most fluent speakers today hear both him being taken and his being taken as correct. That is, people can parse both forms. The first form works by analogy with I helped him learn and the second form works by analogy with I helped his education.

Sentence #2

They are remembered taking me to the zoo.

This is actually correct, just a little unusual. Here, taking is a present participle, not a gerund. Taking me to the zoo modifies they, in the manner of a subject-complement. A comparable sentence might make this clearer:

Football players are usually drawn running with the ball.

In other words, in most pictures of football players, the football player is running with the ball.

The reason your example sentence is unusual isn't because of the grammar, it's because it's a little hard to imagine a situation where people would be remembered that way. But it's certainly possible and the sentence can be understood. Perhaps someone took a photograph of "them" while they were taking you to the zoo, they died long ago, and since then, many people have seen this photograph.

Taking your last example first: do you understand the gerund phrase? This is a phrase that begins with a gerund, where the -ing form of the verb acts like a noun.

I enjoy (sitting on my roof)

I remember (seeing the Beatles in concert)

In the same way "knowing ..." is a gerund phrase that is the object of "do you remember (X)?"

Do you remember (knowing how to play the guitar)?

Take this and change it to the present perfect, which indicates a life experience:

(I have known how to play the guitar.) Do you remember having known how to play the guitar?

(I have worked with Java) Do you recall having worked with Java?

In a similar way, the "to be" verb "is" can be used as a gerund:

(It is open) I remember it being open.

(It is defined in the manual) It is no less confusing, even after being defined in the manual.

This syntax is convoluted and I personally don't recommend using it, if possible. Most of these examples are better when written with simpler grammar:

There is a difference between the practical knowledge you get from working in Java, and the knowledge required to pass this test.

Have you known more than one Amit ... in your life?