This is an answer about the trains and the strange behaviour of the trains website.
The Israeli Rail website regards the trains that leave after midnight as train on the next day, and will not go to them when pressing on the "later" button when looking at train times, for example, the times for trains tonight (Nov, 8th, 2016) are:
In order to see the trains after midnight, you must make a new search for early the next day:
So you can see that there is a train leaving at 23:35 and at 00:50. On normal weekdays, trains from Ben Gurion Airport to Tel Aviv are available all night, and those trains continue north to Nahariya but stop only on the fallowing stations:
- Tel Aviv-Savidor Center
- Haifa-Hof HaKarmel (Razi`el)
- Haifa Center-HaShmona
- Kiryat Motzkin
On Fridays, trains stop in the afternoon, on this Friday the last train will leave at 14:55, the times change during the summer and winter.
On Saturday nights the train work with a slightly different time schedule, for example this coming Saturday trains leave at 23:25 and 00:50.
Holidays are like Saturdays, and the day before the holiday is like a Friday.