How do I make the tick of the axis minimum fall exactly on the other axis?

I set the minimum of the left axis to be zero, but the blown-up view shows that the tick Zero is a little above the bottom axis. Can I make the tick Zero fall exactly on the bottom axis? (Actually the tick Zero will then become invisible, being aligned perfectly with the bottom axis.)

From your picture, it looks like you may have the 'Axis Standoff' checked (on). Go into the Axis GUI and see if that is the case.