Why doesn't the fusion process of the sun speed up? Am I correct in saying that the fusion process of the sun is constant, i.e. X amount of fusion happens per day, more or less?
Why does this not speed up, i.e. one fusion event creates energy for two fusion events, etc.?
Does every collision of atom cause a fusion event, or is the probability small for a fusion event to happen thus it's not a runaway reaction?
I have heard that the probability for fusion event to happen is only 1 in 1012 for every collision.
Am I correct in saying that the fusion process of the sun is constant, i.e. X amount of fusion happens per day, more or less? 


Yes, at least over human timescales. You could reasonably expect the fusion rate within the sun to be the same today as a few thousand years ago or into the future, give or take some small fraction.


  Why does this not speed up, i.e. one fusion event creates energy for two fusion events, etc.? 


The energy released by fusion is quickly distributed as thermal energy in the centre of the sun, and the temperature difference between surface (around 6000K) and centre (estimated 15 million K) drives an energy flow from hot to cold.


  Does every collision of atom cause a fusion event, or is the probability small for a fusion event to happen thus it's not a runaway reaction? 


Fusion in the sun is not a runaway nuclear reaction (like a critical mass of uranium in a fission reaction).

It is possible in theory to have runaway fusion events, but the pressure and temperature for these to happen are not approached in the core of the sun. For stable stars like the sun, the forces and energy flows are in equilibrium - if the core grew slightly hotter, then the pressure would increase and the star expand slightly against the force of gravity to compensate. Interesting things happen when stars fall out of equilibrium and runaway fusion ignition can happen in some scenarios. 

In addition, this equilibrium point moves during the lifetime of a star as its mix of elements changes due to fusion. This is predictable for many stars and forms the basis of the main sequence stars in the Hertzsprung-Russell diagram


  I have heard that the probability for fusion event to happen is only 1 in 10^12 for every collision


I don't know the accuracy of that, but it seems reasonable. The definition of "collision" becomes somewhat arbitrary in such a hot dense environment. If you only include approaches close enough to make the strong nuclear force dominate the interaction, the ratio could be higher.

Another fact that I found interesting in the same area is that the power density from fusion - i.e. the Watts per cubic metre of substance - in the sun is roughly the same as that found in a typical compost heap. It is a very different environment to the inside of a fusion reactor experiment or a fusion bomb, which have much higher power densities.