Athens to Cairo

What you want to do is a extended version of the classic Istanbul-Cairo overland route about which there is way more than decent info on the Net, there was even a LP guidebook dedicated to this route. Besides the Syrian civil war which makes Syria impassable and the the uninterrupted overland route between Cairo and Athens impossible, there are not really stumbling blocks. Visiting Israel can e a little complicated if you want to avoid an Israeli stamp in your passport but no country in your list has an issue with the Israeli stamp, assuming you do not intend entering Syria. If you want to include Lebanon which would also be very difficult without flying given the Syrian war this can be an issue but there are ways to get around it which you can easily find out with a quick search.

What you say you want to do is indeed a lot and not very feasible in 3-4 weeks. But you also say you have done lots of travelling before so I assume you are able to make such feasibility analyses. You can see some sights in each country by doing a lot of flying, if this is your intention then look at Pegasus Airlines which has flights between Tel-Avis and Istanbul and Istanbul and Athens.

You say you want to see whatever is left from the ancient world but do not think whatever is in ruins is worth a visit. Let us know if you know sights from ancient world that are not in ruins.

Agree completely with #2

Visa is not really an issue for that route. Due to the conflict in Syria you can't travel there so would have to fly Turkey to Jordan.

Even if you just did the main places you listed:

Athens, Istanbul, Petra, Jerusalem, Tel-Aviv, Cairo, Alexandria and Luxor

At 3 days per place this works out to 24 days without taking into acount transport which is going to take the best part of a day, or maybe even longer to get from a-b. You would also be missing out plenty of interesting places along the way.

For that reason I would suggest with your timeframe maybe just doing Greece and Turkey (even with 4 weeks you would barely scratch the surface) or Jordan Israel and Egypt

Rhodes, Olympia, Troy, Ephese and Halicarnasse as well but I presume that these places are in ruins anyway and not worth the effort on such a small trip

Well they are ruins, but I wouldn't say they aren't worth the effort.

I would suggest with your timeframe maybe just doing Greece and Turkey ... or Jordan Israel and Egypt

This makes lots of sense. Though even with 4 weeks you would be rushing Jordan, Israel and Egypt.

You could do it the other way around..its still a lot for 4 weeks but not impossible.

Start in Luxor (or in Alexandria...whichever interests you more)..overland train to Cairo (Giza).... overland and boat to Jordan (Petra)...overland to Jerusalem and Tel Aviv.

Fly to Istanbul from there (Pegasus has cheap flights).Then fly to Athens and finish there.

People ahead of me have said most of it - but I'll just say: Don't bother with Troy.
Ephesus, OTOH, is wonderful!
I haven't been to the other places, apart from Petra - which needs two full days as a minimum, I'd say.