No need to worry about the onward ticket, Israeli officials will not prevent you from entering Israel if you don't have one as long as you can explain how and when you plan to leave the country.
For Petra, I would strongly, very strongly advise you to plan two full days to discover the site. The bare minimum would be one full day. It's gigantic and one of the most beautiful things on earth. Combining Petra with a night in the Wadi Rum, as suggested above, is a good idea.
In Israel there are tons of things to see so it's a matter of taste and everyone will have different ideas for you. On my side, I would say:
- Visit Jerusalem's Old City, including the Haram Al Sharif / Temple Mount, the underground tunnel at the Western Wall, the view from the rooftop terrace at the Austrian Hospice (if you can book a room there)
- Visit the Crusader city of Akko (Saint Jean d'Acre) on the sea, sleep in a hotel inside the Walls and have dinner at seafood restaurant Uri Buri on the port, which I consider to be Israel's best restaurant (Akko's Old City is populated by Palestinians citizens of Israel, with Uri Buri being the only Israeli restaurant there, famous for its creative chef).
- Visit the Kabbalist city of Safed in Galilee, have lunch at Rafa's restaurant inside the Artist colony at Rosh Pina village, see the view at the Mount of Beatitudes over the Tiberiade Lake
- Visit Tel Aviv (Neve Tzedek village, and the bars around Rotshchild and Lilenblum), visit Jaffa (the Palestinian old neighborhood with a flea market and gorgeous seaviews)
- Stop at the Dead Sea (can also be done on the Jordanian side, which personnally I consider to be more beautiful)
On the Palestinian side you can visit Bethleem of course, but also Hebron's Old City (take a Palestinian taxi in East Jerusalem to drive you there, it's easier than the bus). Ramallah has a fascinating nightlife if you like to go out. One of of the most beautiful hikes you can do in the Middle-East is the famous Wadi Qelt hike near Jericho, which stops at Saint Georges Monastery, but if you travel during summer it will be extremely hot and should be done with a guide.
- Visit Bethleem and Hebron's Old City