In Israel, do they generally go by the Hebrew or Gregorian calendar (ie 5767 or 2007)in daily life?

Answer 1

Most secular people go by the Gregorian calendar for day-to-day interactions and transactions.
Ultra religious people use the Hebrew calendar more heavily but still have to incorporate some usage of the Gregorian calendar into their daily lives, as they are a minority.
Many institutions, bodies, government ministries, websites, official documents, etc. display both dates simultaneously.
Many calendars in use show both dates.
Since all Jewish holidays are set according to the Hebrew calendar (which is a lunar one btw), people are aware of the Hebrew date to some extent, especially around the holidays.

Website for converting between the two formats:
http://betmishpachah.org/hebrewdate/index.html

Answer 2

Gregorian