I've just started learning Esperanto, and I came across the use of these words as conjunctions.
However, if word order is mostly free, and these question words can be used as conjunctions, isn't it ambiguous which one is the question and which one is the conjunction?
For example:
"Kie estas vi, kiam ŝi estis mortigata?"
If I didn't make a mistake, this means "where were you, when she was being killed?"
However, because the sentence order could be either way, this could also mean, "Where you were, when was she being killed?" (Which, I suppose, could imply that you live in a place with a different time zone, so while she was being killed, it was a different time for you.)

The only solution is either picking up which is which by context, or having a particular word order be correct.

Or, there's something I don't know of yet. How does this work?

Kie estas vi? Kiam ŝi estis mortigiata?

Don't make things too complicated.

thyrolf:Kie estas vi? Kiam ŝi estis mortigiata?

Don't make things too complicated.

Okay, that's not complicated, but wrong.

Yumemitai:I've just started learning Esperanto, and I came across the use of these words as conjunctions.
However, if word order is mostly free, and these question words can be used as conjunctions, isn't it ambiguous which one is the question and which one is the conjunction?
For example:
"Kie estas vi, kiam ŝi estis mortigata?"
If I didn't make a mistake, this means "where were you, when she was being killed?"
However, because the sentence order could be either way, this could also mean, "Where you were, when was she being killed?" (Which, I suppose, could imply that you live in a place with a different time zone, so while she was being killed, it was a different time for you.)

The only solution is either picking up which is which by context, or having a particular word order be correct.

Or, there's something I don't know of yet. How does this work?

I would just say:
Kie vi estis, kiam ŝi mortis?
Or, if you want to emphasize that she was killed:
Kie vi estis, kiam ŝi estis mortigata?

You can use estis in both parts of the sentence, because the word kiam already shows that both actions took place at the same time.

"Kie vi estis" and "Ki estis vi" have the same meaning. There is only a small difference of emphasis. "Kie" has to be the first word. But the next word after it has more emphasis:

"Kie vi estis?" vi is more important.

Yumemitai:
If I didn't make a mistake, this means "where were you, when she was being killed?"
However, because the sentence order could be either way, this could also mean, "Where you were, when was she being killed?" (Which, I suppose, could imply that you live in a place with a different time zone, so while she was being killed, it was a different time for you.)

Well, actually Esperanto works also with context and clearness from the speakers, so there is not such a problem and (if I understand you well) the second question should be formulated as Kioma horo estis tie, kie vi estis, kiam ŝi estis mortigata? or perhaps Tie, kie vi estis, kiam ŝi estis mortigata? This second option is more literal to your question in English.

Anyway in Esperanto, misunderstoods can also to happen.

Ĝis. Novatago (blogo / 7 + 1)

Yumemitai,

There are a couple of things I don't understand in your original post.

What does "were being killed" mean? That the killing took a long time?

If yes, then you can use estis mortigata or oni mortigadis. The latter form, the oni passive is to be preffered because of its simplicity.

If not, then simple oni mortigis is enough.

English speakers tend to overuse different kinds of esti + participo constructions.

I don't get the timezone aspect. Anyway combining what Horsto and Novatago wrote with what I said above:

• Kie vi estis, kiam ŝi mortis? : Where were you, when she died?
  
• Kie vi estis, kiam oni mortigis ŝin? : Where were you, when they killed her?
  
• Kioma horo estis tie, kie vi estis, kiam oni mortigis ŝin? : What time was it there where you were, when they killed her?

Yumemitai, in English we can easily distinguish between the question use and the qualification use since we invert the order in the question use.

Where were you (inversion) when she was (normal order) killed?

However in Esperanto the inversion doesn't create a question. So if you say Kiam oni mortigis Kennedy .. and pause before continuing but intending to continue with a question, an eager interlocutor might jump in with 1963 rather than answer your actual intended question which you hadn't yet completed..

"Where you were, when was she being killed?" is an incomplete sentence, so it would not be interpreted that way.

The k words (kie, kiam, kio, and so on) tend to indicate the sentences are questions when they are the first words in the sentences, and tend to function as conjunctions when they are buried inside the sentences (especially immediately following commas). Word order *does* matter in Esperanto, it just doesn't matter *as much* as in English.

Now, this is not a hard and fast rule, it's true. "Kiam", for example, may start a sentence that isn't a question, but rather as part of a statement meaning "while something was happening". "Kiam mi iris al la vendejo, mi bla bla bla...". That isn't a a question, even though it starts with a question word (although it would probably be better expressed as "Irante al la vendejo, mi...", it is common to hear the kiam form). But it's pretty easy to tell that from the construction of the rest of the sentence. Pay attention to context.

In fact, in sentences of the form "Kiam mi iris al la vendejo, mi bla bla bla..." , there's actually a word being elided (left out because it's implied). That word is "tiam". The sentence would be fully expressed if stated as "Kiam mi iris al la vendejo, tiam mi bla bla bla..."., as that firmly indicates that the sentence is not a question but rather a statement. But more frequently than not we drop "tiam" altogether with the idea that it will just be understood as being there from the context of the rest of the sentence. Many if not all of the k words work this way, with the paired t words understood as just being there. "Being Colloquial In Esperanto", a book definitely worth checking out (especially because I believe it's free nowadays), goes into this in almost painful detail.

RiotNrrd:"Kiam", for example, may start a sentence that isn't a question, but rather as part of a statement meaning "while something was happening". "Kiam mi iris al la vendejo, mi bla bla bla...". That isn't a a question, even though it starts with a question word (although it would probably be better expressed as "Irante al la vendejo, mi...", it is common to hear the kiam form). But it's pretty easy to tell that from the construction of the rest of the sentence. Pay attention to context.

I would argue that a kiam subclause is more easier understood by many and thereby more along the fundamental (pun intended) idea of Esperanto than adverbial participles.

You're right that it's pretty easy to tell that whether it is a question.

Yumemitai,

Compare
 

• Kiam mi iris al la vendejo?
  only one clause in the sentence, the clause begins with an irrogative, this must be question
   
  
• Kiam mi iris al la vendejo, ĉu mi bla bla bla…?
  two clauses in the sentence, both begin with an irrogative, i.e. this must be question, ĉu takes precedence in such case
   
  
• Kiam mi iris al la vendejo, ▁ mi bla bla bla…
  two clauses in the sentence, only one contains an irrogative (*), the irrogative must be a conjunction.

*: There are words like se, kvankam which can make things a little bit harder.

• Kiel vi povos iri al la vendejo, se mi bla bla bla…?
  
• Kial vi iris al la vendejo, kvankam mi bla bla bla…?

but here helps the fact that such words never begin a main clause, i.e. the ki-clauses are the main clauses and so these are questions.

Oh, I'm not saying people shouldn't use the kiam form. There even is a subtle difference between the adverbial form ("while I was...") and the kiam form ("when I was..."), although in practice that difference doesn't often make a lot of difference. Nevertheless, there is a hair there that can be split, if people want to. *I* like the adverbial form because it's slightly more compact, but that preference is just a stylistic choice on my part. Both constructions are totally fine, mean "more or less" the same thing, and either can be used with wild abandon.