Mostly the sentences look good. I'm not experienced enough with composition to offer suggestions on word choice here, so I'll restrict my comments to purely grammatical points.
εκ τoῦ ἵππoυ εξῆλθoν oἱ Λακωνικoί καί τoύς Tρoιᾱνoυς έσφασαν.
Perhaps I'm being obtuse here, but I'm not sure what verb you're using to get έσφασαν
ἱσπᾱνικoί τήν μεγάλην πóλιν χρῡσῆν εζήτoυν.
I assume you want to say the "great golden city", correct? χρῡσῆν is in predicative position here, as though you were saying "(they thought) the great city was of gold". You want something like τήν μεγάλην καὶ χρῡσῆν πóλιν or τήν πóλιν τήν μεγάλην καὶ χρῡσῆν. (Greek doesn't distinguish between adjectives of size/quality and other kinds of adjectives the way Latin does -- they all can be placed in the same positions relative to the noun. What is important is the placement of the article.)
*I had to put the aspiration mark on the eta rather than the upsilon because combining it with the tilde just looked too cluttered and unreadable.
Just so you know, according to typographical conventions ἡῦρεν (with the aspiration and accent on different vowels) would indicate that the word is to be pronounced as three syllables (not as two). In this case it doesn't necessarily prevent understanding, but there's a few cases where the distinction is crucial. For example -- αὐτή would mean "she" or "herself"; however, ἀϋτή means "a call or cry". The placement of the accent over the alpha in the second word tells you that it's not a diphthong (the two dots over the upsilon also indicate that the vowel sounds are separate, but because the aspiration already makes that clear, many editors will omit it).