Mi respondis trankvile, ke mi ne eraris.

A simple breakdown is:

• Mi = the subject, I
• respondis = past tense of respondi, answered / replied
• trankvile = an adverb, calmly
• ke = that, introducing a subordinate clause
• mi = the subject of the subordinate clause, again I
• ne = not
• eraris = past tense of erari, was mistaken / made a mistake

So the sentence has a main clause plus a ke-clause.

Because it modifies a verb, not a noun.

In Esperanto:

• -a = adjective
• -e = adverb

Here trankvile describes how the speaker answered, so it must be an adverb.
If you used trankvila, that would describe a noun, such as mi estis trankvila.

Ke introduces a subordinate clause, like English that.

So:

• main clause: Mi respondis trankvile
• subordinate clause: ke mi ne eraris

It connects the answer/reply to its content. In other words, it shows what was said in the reply.

In Esperanto, subordinate clauses are commonly separated with commas, and that includes clauses introduced by ke.

So the comma helps show the structure:

• first, the main statement
• then, the content clause

This is very normal Esperanto punctuation.

Because the second clause needs its own subject.

Esperanto verbs do not change according to person:

• mi eraras
• vi eraras
• li eraras

Since the verb ending does not tell you who the subject is, Esperanto usually states the subject explicitly in each clause when needed. So ke mi ne eraris needs its own mi.

In Esperanto, ne normally goes directly before the word or phrase being negated. Here it negates the verb:

• mi ne eraris = I did not make a mistake / I was not mistaken

This is the most natural position.
Putting ne elsewhere can change the focus or sound unusual.

Because the speaker is presenting the non-mistake as something in the past.

A very important point for English speakers: Esperanto does not use tense in subordinate clauses the same way English often does with backshifting. The tense in Esperanto usually reflects the actual time of the action or state.

So:

• ke mi ne eraris = that I was not mistaken / had not made a mistake
• ke mi ne eraras would mean that I am not mistaken now

Both are possible in the right context, but they do not mean the same thing.

Erari means to be mistaken, to make a mistake, or to be wrong.

Which English translation fits best depends on context. In this sentence, it could naturally correspond to:

• I was not mistaken
• I had not made a mistake
• I was not wrong

Esperanto often uses one verb where English might choose several different expressions.

Yes. Esperanto word order is fairly flexible.

For example, these are also possible:

• Mi trankvile respondis, ke mi ne eraris.
• Trankvile mi respondis, ke mi ne eraris.

The original order is perfectly natural, but moving trankvile can slightly change the emphasis.
What usually should stay the same is that ne remains right before the thing it negates.

Yes. You can add that if needed.

For example:

• Mi respondis al li trankvile, ke mi ne eraris.
• Mi respondis al ŝi trankvile, ke mi ne eraris.

With respondi, the person answered is often introduced with al.
If that information is not important, it can simply be left out, as in the original sentence.

Because respondi means to answer / reply, while diri just means to say.

So respondis tells you that this statement was made as a response to something.
If you said Mi diris trankvile, ke mi ne eraris, it would mean I said calmly that I was not mistaken, but it would not specifically imply that this was an answer to someone.

Yes, completely natural.

It uses very standard Esperanto grammar:

• normal past tense in -is
• normal adverb in -e
• a standard ke-clause
• normal placement of ne
• normal punctuation

So this is a good model sentence for everyday Esperanto.