Sunday-Thursday: Most businesses open at 8-9am, close at 6-7pm. Banks are often open only half the day. Supermarkets and large pharmacies stay open until 9-10. Restaurants are open until 10 and often much later.
Friday: A half-day, businesses close around 3pm (in the old city around noon).
Saturday: All businesses in the Jewish neighborhoods are closed (except a small minority of restaurants and some gas stations). Most restaurants, and some (but not all) supermarkets and pharmacies, open around 8:30-9pm. Hotel services (including dining) and museums operate as usual. Businesses in east Jerusalem and the old city are generally open.
In Tel-Aviv restaurants are mostly open on the weekend.
August is dry and hot (30+ degrees C is not unusual in the mid-day). In the evening things cool down to a comfortable and sometimes even chilly temperature. Most people wear a short-sleeve shirt, light pants or shorts in the day, and possibly a sweater for after dark. For more information see here.
There is no public transport from 3pm Friday to 9pm Saturday! (except taxis and airport shuttles).
The main languages are Hebrew and Arabic, but essentially everybody speaks English.
Behavior that may seem aggressive to you is probably just self expression.
Be respectful in holy places and dress appropriately.
Tipping in Restaurants is customary, 10%-15% is the usual range. Cash only.
Jerusalem has a very low crime rate (though like any city some neighborhoods are less inviting). In the university and center areas there is no problem walking at night. If in doubt, feel free to consult the locals.