This reading I'm working through has a lot of sentences structured like <predicate phrase> <subject/topic noun>. My understanding is that these are actual sentence fragments, comprised of a single noun phrase where the 'subject' is being modified by a relative clause (I think I've read that certain styles of poetry favor a lot of sentence fragments like this, but I have no idea if this is accurate).

I'm not really having issues understanding the meaning of these, in terms of who's doing what, but I'm wondering two things:

1. Should I be interpreting these as complete sentences, which have just been inverted for whatever reason, instead of fragmentary noun phrases?

2. In either case, any suggestions on how to translate these into English in a way that doesn't sound stilted? What kind of tone is being conveyed by this structure?

In a sense, you can interpret a lot of Japanese sentences as acting like they have a copula even if it's ommited, so interpretimg all of these as sentence fragments might me going too far. I interpret some such sentences as fragments, especially if they sound stilted. Examples would be nice though.

どこかにいる子供。。。 最後に見た所？

I'm reading this more literally as "The child, who is somewhere... The place where you saw him last?" and would probably rewrite it in English like "The child is somewhere... Is this the place you saw him last?"

I'm taking the second sentence as having an implicit copula, but I'm not sure about the first one.

To me, that reads more like "The child who is somewhere... Where did you last see him?", though I could be wrong if context says otherwise. In this case, the topic is the child while the subject is 「最後に見た所？」 "place where last seen". In this case, the interrogative is dropped, but to me, that seems more likely, since it can be implied through the tone (indicated by the question mark), while a specific location like 'here' would have to be previously established.

As for how to interpret them... technically they're sentence fragments, but they go together, so I guess it's just a written expression of spoken brevity (particle omission, copula omission).
Such abbreviation might suggest that the speaker is either hurried to the point of not caring about proper speech or is familiar with the listener (or otherwise unconcerned with proper speech). At least, that's my understanding of it.
As for translating it to English, you might consider phrasing that captures that same kind of brevity ("The child... where was he?"). It all depends on context, though.

Of course, the problem with my interpretation is that the sentence you provided doesn't show abbreviation in the noun phrases that would fit with such brevity, so... maybe it fits with the 'no need for formality' brevity than the hurried brevity.
If this isn't resolved before I see my tutor again, I'll ask her, but that's a good two weeks from now.

If it's "Where did you last see him" i would expect a は at the end; what's the context of this?

Do you mean something like this (http://allabout.co.jp/gm/gc/450223/)?

新年会や生活リズムの乱れで太りやすい年末年始。

Which could be thought of as:

年末年始は新年会や生活リズムの乱れで太りやすい(もの・時)です。

I see this a lot at the start of articles about some subject x. In this case, it acts like a colon in English. "The New Year's holiday: When New Year's parties and a broken life rhythm mean it's easy to gain weight." Or like two English sentences: "The New Year's holiday. It's a time when New Year's parties and a disrupted life rhythm can spell easy weight gain."

Best not to think too much about the English equivalents and just take it for what it is: a way to introduce a subject with some colorful detail added.

gaiaslastlaugh Wrote:Do you mean something like this (http://allabout.co.jp/gm/gc/450223/)?

新年会や生活リズムの乱れで太りやすい年末年始。

Which could be thought of as:

年末年始は新年会や生活リズムの乱れで太りやすい(もの・時)です。

...

That looks like what I'm seeing. I'll try going through some of the stuff again in light of your suggested translations and see how that works out for me.

Translating the sentence I was looking at as "The child -- they're somewhere [else]... Where is the place you last saw them?" sounds alright in English to me.

Saginaim Wrote:どこかにいる子供。。。 最後に見た所？

I'm reading this more literally as "The child, who is somewhere... The place where you saw him last?" and would probably rewrite it in English like "The child is somewhere... Is this the place you saw him last?"

Hard to say without more context, but it almost sounds as if the first part of the sentence, including the "。。。", is a restatement of what the other person said, kind of talking or wondering out loud. My imagination leads me to this kind of scenario:

Person A: "Officer, I told my child to wait here while I fetched my umbrella from the car, but when I returned my child was gone! He must be around here somewhere!"

Officer: "Hm... a child, [who got lost] somewhere around here... where did you see him last?"