Say I have a sentence that says "遠くに町の灯が瞬いていた", that is, "the lights of the town were twinkling in the distance". Is my understanding correct, that if I instead want to construct the clause "the lights of the town (that are) twinkling in the distance", I would then just move the subject to the end of the sentence (and change the grammatical tense), as I have done below:
遠くに町の灯が瞬いていた
Becomes:
遠くに瞬いている街の灯
Or do I, by doing this, break the grammar somewhere I haven't been able to tell?
Generally yes, that is how one can construct a noun phrase from a sentence although tense/aspect may require different considerations for each verb.
猫がにゃーにゃー鳴いていた The cats were meowing
にゃーにゃー鳴いている猫 The meowing cats
Consider
太郎は花子が好きだった Taro liked Hanako
Applying the same construction, we get
花子が好きな太郎
This is ambiguous between "Taro who likes Hanako" and "Taro whom Hanako likes".
Another example:
花瓶が割れた The vase was broken
割れる花瓶 is not possible (or means an object that works as a vase and has separable parts). "A broken vase" is always 割れた花瓶.
遠くに瞬いている街の灯 is a perfectly correct phrase that means "the lights of the town twinkling in the distance". But note that, strictly speaking, this is not a clause, but a (noun) phrase that contains a (relative) clause.
You may be worried that 瞬いている might modify only 街 (i.e., "the town that is twinkling"), but you don't have to worry about this. When we look at the English version, we understand "twinkling" modifies not just "the town" but "the lights of the town" even without thinking, and that's because we know a town itself doesn't twinkle. Likewise, also in Japanese, everyone understands 瞬いている modifies not just 街 but 街の灯 as a whole.
Related: Are Japanese modifiers "greedy", "anti-greedy", or do they mean whatever people choose them to mean?
