Voltage Follower - OPAMP
1) The opamp output doesn't follow gain*input_differential immediately, it takes some time, due to both bandwidth and slew rate limitations. That delay gives the time for the amplifier to find the 'correct' voltage.

What happens if it overshoots again and again? This is a thing in any amplifier circuit, stability. Most opamps take care of it by hiding the details from you, and using a dominant pole to give you unity gain stability. This means that most noobs can just wire an opamp and it will  be stable. The cost of this ease of use is an amplifier that's slower than it could be. Professionals often use uncompensated opamps, which are faster, but need stability thinking about. It's still possible to get a 'unity gain stable' opamp to be unstable accidentally, but it usually takes a more complicated circuit than a follower to do it.

2) In an ideal opamp, gain is infinite. In a real opamp, gain is very, very large. For a given output, the input will settle to output/gain. As gain approaches infinity, the input approaches zero, which we can approximate to zero with little error. Usually the DC error we get from offsets exceeds the error due to finite gain, so it's rarely the error we have to worry about first. If we compare two opamps with gains of say 100,000 and 1,000,000 each producing 1 V output, then one will have 10 &mu;V input error, the other 1 &mu;V. However, they may have several mV input error due to input offset voltage, which then changes at several &mu;V per degree change in temperature.

3) Output source current comes from the +ve rail, output sink current goes to the -ve rail. There is a maximum for both of these currents. In keeping with ease of use, most opamps will actively limit how much current they will source or sink in the case of an output short circuit, so that they will survive brief accidents with the trailing 'scope probe ground. The specifications for any given amplifier will detail how much they are expected to produce while staying within specification, and what they will limit at.