Airport shops are usually expensive even without taxes. But do have a look when you're in the airport (either in Lebanon or there), you might get lucky. Try not to miss your airplane though, I once almost missed my flight because I was too busy checking out cameras.
If your parents are not travelling then they will not be able to get to the duty free shop area in the airport, so they can't help.
Airport shops are duty free, so the way I understand it is you don't pay VAT therefore you cannot ask for VAT refund. I am not sure though.
Duty free shops will ask for your ticket but not your passport, I think, so your nationality would not matter. Again, have a look and ask when you're there.
Your best bet for buying a cheap phone is ordering it on ebay, Amazon, or such - you will need to know an address of someone to deliver it to, or if you're staying a long time (a week or more) at the same hotel you can use the hotel address. I would not try this if I'm staying a few days at the same hotel because of the variance in delivery delays, it can arrive 2 days (or more) late or 2 days early especially if ordering from another country.
You could also research online to find good shops there and go get one once you're there. If you're going to a big city then you might get luck and find a good deal on some site for a shop there.
Then on your way back you can apply for VAT refund, however I'm not sure that you can get refund for stuff bought online. I think you just need the invoice and your passport and visa. Look up the details online.
You can get VAT refund when you buy things on a touristic visit. So you would be a tourist in Italy.
If you want to cheat and buy the phone in Lebanon and try to get a 11% VAT refund by pretending that you're on tourism here, I don't know how it works, I don't know much about VAT refund here, and I would not bother trying this personally.
It will be difficult to argue that you're on tourism in Lebanon when you are showing a Lebanese passport.
This is my advice based on what I know.
xazbrat wrotethe phones are cheaper here than in the EU
This is true, except if you are looking for a specific model that is rare then it will probably be more expensive here.
So if the phone that you want is generally available in Lebanon, better buy it here then.