I am not talking about how it is written, but how it is pronounced. Before back vowels, like o or u, it seems to sound like a ZH, the sound of the letter s in the English word "measure." Sometimes here before front vowels (like i or e) it sounds maybe like KH, the χ sound, or even maybe like the voiced TH in the English "the" or the Greek letter δ. I'm sure most of this is just the recording coming out of my speakers, but clearly the ζ is making a sound that is not like the English Z in "zebra."
You use the pronoun when you want to emphasize who did the action, as opposed by someone else. I am working while you are sitting and having fun!=Εγώ δουλεύω ενώ εσείς κάθεστε και διασκεδάζετε! But in any other case the pronoun is omitted. I am eating=Τρώω, I am walking=Περπατάω etc
Is Έχουμε pronunciation: ehume or ekume?
The χ in this case is pronounced like the "ch" in "loch", if you have a Scottish accent. It's not an English sound for many speakers of English.
You can find the sound in German, Hebrew, Russian, Arabic, or other languages, though.
The older form is ἡμεῖς: https://en.wiktionary.org/wiki/%E1%BC%A1%CE%BC%CE%B5%E1%BF%96%CF%82