So, to make sure that I don't make any big mistakes, I sent an email to Hertz Israel to ask about this, And they already answered (very quickly). The answer was:
"We do have a key box in Eilat branch- So if the rental is Sunday to Sunday it's ok."
I will probably do that.
I have taken the bus a few times, but never with kids. They'll be able to see much more if we can stop along the way.
Thanks for your suggestions, Ditto!