Generally, you’re on the right track, but there are a fair number of mistakes. Here’s how it scans, using your notation:
κυπρογε|νῆ Κυθέ|ρειαν … ἀ|είσομαι, | ἥτε βρο|τοῖσι
μείλιχα | δῶρα δί|δωσιν, … ἐφ’|ἱμερ|τῷ δὲ προ|σώπῳ
αἰεὶ | μειδιά|ει … καὶ ἐφ’|ἱμερ|τὸν θέει | ἄνθος.
χαῖρε, θε|ά, Σαλα|μῖνος … ἐ|υκτιμέ|νης μεδέ|ουσα
εἰναλί|ης τε Κύ|πρου … δὸς | δ’ ἱμερό|εσσαν ἀ|οιδήν.
αὐτὰρ ἐ|γὼ καὶ | σεῖο … καὶ | ἄλλης | μνήσομ’ ἀ|οιδῆς.
No trochees! Only dactyls and spondees. The last syllable of the line is stretched out and treated as long, even if it would otherwise be short, so that the last foot in a line is always a spondee. In other positions, trochees are foreign to the hexameter, with a very few, very rare exceptions, and when they occur, they are scanned as spondees. If you think you see a trochee, you’re almost certainly wrong and should go back and re-work the scansion.
The strong pause in the middle of the verse – the “caesura” – usually comes in the third foot, either after the first long syllable (“masculine” caesura) or, if the third foot is a dactyl and not a spondee, the caesura can come after the first short syllable (“feminine” caesura). Feminine caesuras generally predominate in Homeric and archaic hexameters. Lines 1, 2 4 and 6 above have feminine caesuras in the third foot; the others have masculine caesuras in the third foot. Sometimes there is a caesura in the fourth foot after the first long, frequently with another in the second foot, but that doesn’t occur in these verses.
The caesura is a break between words within a foot – it never splits a word, and it always occurs within a foot, never between feet. Thus, your scansion αἰεὶ μει|διά … | ει καὶ ἐφ’ | ἱμερ … | τὸν θέει | ἄνθος is wrong on several counts. The caesura is the most important feature of the hexameter–more important than the division of the hexameter into metrical feet. The two halves of the verse (or in some cases, three sections) are separate metrical segments. If you’re going to sing hexameters (and even if you aren’t), you should get to recognize the caesuras and be very attentive to them.
There are no instances of syezesis in these verses; your scansion is wrong where you’ve indicated synezesis. There are, however, instances of correption – a long vowel or diphthong at the end of a word is usually treated as short if followed by a vowel beginning the next word. In the third line καὶ is short before ἐφ’, and -ει is short before ἄν-. θέ-ει is two syllables, both treated as short. In the sixth line, καὶ is short before ἄλ-. σεῖο … καὶ is a dactyl – three syllables, with a feminine caesura.
Note that the first syllable of κυπ-ρογενῆ in line 1 is treated as long, but the first syllable of Κύ-πρου in line 5 is treated as short. The vowel υ is short, but the consonant cluster πρ (among other combinations of obstruent + liquid) can be treated as either lengthening a short vowel that precedes it, or not.
Hope this helps.