You are asking good questions...
> 1. He says water reduces reactivity but steam increases it. How does this really work?
You're first sentence is really saying the same thing. In the case of the Chernobyl reactors, water acts as a neutron absorber. if you add more water, you absorb more neutrons and reactivity decreases. If the water boils, you have less water, so reactivity increases. Why this matters is that when the power increases, the reactivity increases, and this is a really bad design feature (called a positive reactivity coefficient).
Another thing to remember is that neutron moderation is performed with graphite blocks, which are stationary at all power levels.
Contrast this to light water reactors (LWR's) where the water is both the coolant and the moderator. When the power increases, the water density decreases, which decreases the moderation as well as the coolant. Therefore, LWR's have a negative power coefficient. When the power increases, the reactivity decreases.
> 2. He says heat reduces reactivity. (Completely hypothesizing here, is it something to do with phonons interacting with neutrons? I might be out of context though, if so please correct.)
This is something called the "Doppler effect". In the figure in comment:2, you can see "spikes" in the cross sections. These spikes are called resonances. When the temperature increases, the resonances increase in width, and absorption increases. This causes a decrease in reactivity.
So when the power goes up, the temperature goes up, and the reactivity decreases due to Doppler. This is also a negative reactivity coefficient and another very important safety feature.
The Doppler effect occurs in every reactor that has uranium or plutonium fuel (i.e. every reactor)> 3. The deadly flaw about the rods: their graphite tips. He says graphite increases reactivity. But isn't it used for "moderating" purposes in the reactors all over the globe? An elaborate explanation I would appreciate.
Yes, you are correct. Graphite is a moderator. The key here is that the graphite is located on the tip of the control rod. Therefore, when you insert a control rod, you are inserting graphite and increasing reactivity. This is a bad thing, when you insert a control rod you expect that the reactivity will go down, not go up.
The graphite tips are complicated because the total reactivity depends on where the graphite tip is, and where the absorber regions are located relative the core. Almost always, the control rod is located where the neutron absorber has a much bigger affect than the tip, so the control rod is always negative reactivity. However, in the accident, they had the control rods withdrawn too much and the tip was located at the very top of the core and the absorber was completely withdrawn. Therefore, when the initially inserted the control rod back in the core, the tip was dominate and positive reactivity was inserted. This is what initiated the accident, but it was due to the reactor being in a very unsafe initial condition. The operators had to do a lot of things wrong to get the reactor into this condition.