I worked in Israel for 9 months one time... I left @ 6-7 years ago. I had no problem crossing Eilat and caught a cab to Sharm. Fee for the cab was not much as I remember but the music was really shrilling and reminded me of a bad Chinese opera!!! Driving at 3am through the mountain passes without headlight on can cause a moment or three of worry! Bandits as he said!!! I would do it again but only in the daylight...!
Flying from Israel to Egypt (or anywhere else for that matter) is a true test of patients!!! No if's, and's or but's about it... It can take several to many hours to get through security!!!
Personally, I would cab from Eilat it if you had the time to Sharm (catch Dahab on the way), fly from Sharm to Cairo, or if you intend to hit Lexor, see if a ferry crosses from Sharm to Port Safaga, hit Lexor and cab it north to Cairo and then fly into Israel. I loved Israel and would go back in a moment and I didn't even dive at the time... Egypt is very poor, not India poor but I loved India and have little to say about Egypt. Cairo was good for a day or two; seeing the Spinx, Pyramids, Museum, Citadel, and the City of the dead (really poor area so be careful)... Tour Guides will hit you up but expect to pay twice what they say as they expect equal the cost in tip...
I did not go to Luxor so I can not comment but I have been to alot of places and Cairo did little for me other than what I mentioned above to see. I am sure others will suggest many other places to go in or around Cairo but for me, it was good to see what I did see but we spent a week in Cairo and that was probably 4 to 5 days to long... You may love it there but I would enjoy the other areas in Egypt and 2 or 3 days in Cairo... It all depends on what your into...!
If your going to go to other arabic countries, have the Israelies stamp your boarding pass instead of your passport. That stamp will keep you out of some countries and hassels in others... Tell them your a service engineer and travel alot... I was and did and it worked for me... but I also had a passport full of stamps...
Best to you in your journey,
lee
PS When asked, I was from Canada!!! Be careful...