The average depositor is not paying negative rates: http://www.ubs.com/ch/en/swissbank/private/interests.html

There is obviously a limit as to how negative the SNB can push the interbank rate, but if you see it as the "convenience fee" for keeping and moving large amounts of money in the central bank versus trying to hold it and transport it as cash it makes more sense.

It's the rate banks have to pay to deposit Swiss Francs with the Swiss National Bank. Many Swiss Banks however do invest their assets differently (higher risk) and therefore the people in Switzerland with private accounts are not affected directly. This is mostly to avoid foreign banks to invest in Swiss Francs.

Only the interest rates on money borrowed from the swiss national bank (which only other banks can use) are affected. This will keep the interest rate you get for your private account close or equal to zero, but they will not get below that margin for exactly the reason that people will withdraw their money.

It's the rate on bank-to-bank borrowing (basically, the rate banks pay to borrow money from the central bank). But also the franc is like gold - people move their money to the franc for stability and so there's always external demand.

The only run on ATMs* and banks is to get cheap euros as long as still possible.

*ATMs here will give you Swiss Francs or Euros.