Hi friends, my mother and I crossed the border from Jordan to Israel easily. We arrived at the Jordanian border a little before 9am on a Sunday. Passport control on both sides was very light and it took us about 30 minutes to exit the Jordanian side and enter the Israeli side. Despite having stamps from Islamic countries, including the UAE, we did not encounter additional questioning at the Israeli border or at the airports.
We chose to fly Arkia to get from Eilat to Tel Aviv. The process could not have been easier. There were taxis in the parking lot on the Israeli side of the border; the ride took about 20 minutes. We got to Ramon airport far earlier than the flight I had booked. The Arkia agent at the ticketing counter re-booked us onto earlier flight. No change fees, not fare difference, no standby fees - what a pleasant surprise versus changing flights in the US!
We returned to the US via Tel Aviv. This pre-check in security was even speedier than what we experienced at Ramon airport. We did not have any difficulties despite other stamps in our passports and entering Israel via a land crossing.
I think the biggest takeaway for other travelers is the ease of flying from Eilat to Tel Aviv. I had spoken with an Arkia customer service representative over the phone about change fees. The customer service rep told me that if they change your flight over the phone, you might be subject to paying a fare difference and a EUR5 change fee (each one way ticket cost EUR55). The phone rep told me airport agents have more flexibility with respect to re-accommodating passengers (we made our changes at the airport). The airport is bright and airy and the flight was speedy and comfortable.