Welcome to PF;
Those are actually decent questions.
1. Usually the photoelectric effect is cited as pretty definitive for this - a telling trait is that sometimes you get an ejected electron the second you turn the instrument on (i.e. the light wave has not had time to deliver enough energy.)
What distinguishes the (classical) wave from the particle model is that the particles deliver their energy in one sharp hit.
QM accounts for things like interference patterns by having the wave-behavior in the statistics rather than the actual object.
2. You don't need to fire a single photon ... just like you don't need to isolate a single electron to prove that charge is quantized.
http://www.cnrs.fr/Cnrspresse/en25a4.html
"The smallest amount of light that can be considered light" is the definition of "photon".
But I think you are imagining switching a light sourch on and off fast enough that only one photon gets fired out? In fact, the light source is so dim that only individual photons traverse the apparatus in one time ... in a typical experiment, the average distance between successive photons is over a kilometer (i.e. the probability that two or more do so can be made vanishingly small). The experiment is repeated many time to collect statistics (in case two or three did make it through sometimes) and the photon detectors always register light energy in randomly spaced lumps rather than as a smooth delivery ... these days we can use detectors good enough to register a single photon hit.
http://qubit.nist.gov/qiset-PDF/Nam.QISET2004.pdf
There is a semi-classical formulation that tries to account for this.
I understand there is a purely wave theory as well ... also not classical waves.
However, I think it is most useful when you are learning about particle physics to start thinking of everything in terms of fundamental particles: just to make the break with classical waves and particles and any lingering ideas about "duality".
The following lecture series often helps people understand the concepts:
http://vega.org.uk/video/subseries/8
... Richard Feynman on QED and wave-particle duality.