DAPI stains the nucleus but doesn't work well on apoptotic cells. Propidium Iodide (PI) works better if you are trying to analyze all cell cycle phases.
The peaks on the y-axis are the amount of cells, so the high peak at G0/1 is showing that there are lots of cells at this stage in the fell cycle. Along the x-axis is the DNA content so S phase is showing more DNA content than G0/G1, and G2 is showing more DNA content than S and G0/G1
For a fixing protocol where the cells are killed but preserved for imaging. This way you dont have to worry as much about decay of the cells before your analysis. Similar in principle to paraformaldehyde (PFA) treatment of tissue sections or microscope slides.
Presumably you dont have to though if you want to keep your cells alive. Prob put them in serum free media till you need to analyze, then prob swap out your media with PBS for less background signal.
This is also necessary for Propidium Iodide (PI) staining since it is membrane impermeable in live cells. Thermo has some nice non-toxic dyes that distribute throughout the cytoplasm and are membrane permeant to live cells. See CellTrace. They have three kinds, Violet, Yellow, and Deep Red.