The Kastle Meyer Blood test is all about transfer of electrons through reduction and oxidation (called redox for short). If something gains electrons, it is reduced, if something loses electrons, it is oxidized. (Usually something gets oxidized when it reacts with oxygen, which is where the name comes from.) A way to remember this is with a simple mnemonic: LEO the lion says GER (Lose Electron Oxidized, Gain Electron Reduced).
There are a few things going on, so here's a step-by step guide on how it works.
a) Phenolphthalein is colourless when it's reduced (has electrons), and pink when it's oxidized (no electrons).
b) Peroxide can decompose to water if you give it electrons. Normally, it's somewhat stable. (The bottle in your cupboard is still good if you use it next year. It's always a dark bottle because it does react slowly with sunlight.)
c) Normally, peroxide is somewhat stable (see above), but blood causes it to decompose. Specifically, the iron centres in your blood (hemoglobin in your red blood cells) have electrons that it can donate to peroxide.
1) If you start with colourless phenolpthalein and peroxide, nothing happens.
2) However, if there's blood, the blood gives its electrons to the peroxide, and then...
3) the phenolpthalein gives its electrons to the blood.
4) Because the phenolphthalein lost its electrons, it turns pink. (Apparently if you leave it long enough, the oxygen in air will slowly oxidize the phenolphthalein and turn it pink.)
Now that we know how it works, the other questions should be easier to understand. The whole point of adding peroxide is so that the blood loses its electrons and reacts with the phenolpthalein to turn it pink. However, there are other things that also react with peroxide, and other things that will take electrons from phenolphthalein, so you can easily get a false positive.