The 'x' in this case is ξ rather than χ (Θελξιέπεια). That's a 'ksi', which makes an x sound like in English. Also, ιε is not a diphthong so it would be pronounced separately. 'Thelxiepeia' becomes 'thel-ksi-EH-paya' or 'thel-ksi-EH-peeya' if you're going with a modern Greek pronunciation.
Similarly you'd have 'agh-la-OH-pee' in Modern Greek (η is an i/ee sound along with lots of other vowels! - and the stress is on the omicron) and 'pee-see-NOH-ee'.
Of course it's entirely up to you whether you go for a modern Greek or Erasmean pronunication for Ancient Greek names. Neither is more correct, and it just depends what you're more comfortable with.
p.s. slightly OT but I was drawn in by the thread name...κάποιες Έλληνες εδώ; Είμαι Αγγλίδα αλλά μιλάω λίγω Ελληνικά. Πρέπει να έχουμε ένα thread Ελληνικό, ναι;