I am not positive that I am correct, but shouldn't it be "reezee" not "reehee" for rice?
Is the ζ before a front vowel here tending toward a ZH sound or a KH sound?
A ζ is a ζ is a z... always. I'm not sure I understand the question and the TTS sounds pretty clear to me. It is maybe the ς that concerns you?
I think the problem is that the word ρύζι is pronounced differently in the sentence and as a word alone. In the sentence ζ sounds like "z" and as a word it sounds something like "h". Is there a reason for that or is the word version wrong?
There is no reason, the version "r'ihi" is just wrong. "r'izi" is the only correct sound.
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."
That sounds like a definite Z sound. I would guess that what I am hearing, then, is a ZH sound (thicker, as you say), which may be a regionalism of the recording used on Duolingo.
Is there any factor that affects the use (or omission) of the pronoun and which would be most common in every day speech?
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
But wouldn't Εμείς τρώμε and τρώμε have the same meaning? Why is there a need to have Εμείς ?
in my page there are no options to select and typing the word doesnt work. it was not possible to complete the level due to an error in the software.