What is the experimental reason to believe in a Unified Theory? Is there any experimental findings on behalf of the belief that all forces can be unified to a single force? Or the idea has it's inspiration from kind of thought experiment or philosophical belief.
 A: The experimental evidence comes from the running coupling constants.

One important clue that they (couplings)  might all be the same comes from the fact that the coupling constants are not constants at all. Instead, they vary with the energy of the phenomenon in which they are measured. The value of as quoted above is only true for phenomena that occur at around 91 GeV of energy, the mass of the Z particle. Similarly, the coupling constant for the electromagnetic force has been measured to be about 1/137 in studies of atomic phenomena. When it is measured at the higher energies of LEP, however, its value has increased to about 1/128! The value of the coupling constant is said to 'run' with energy, and the constants themselves are usually referred to as running coupling constants. 


You might read the paragraph comparison to experiment on page 29 in this article. 
In this talk the experimentally found running of the strong coupling is seen on a graph on  page 3.
The running of the electromagnetic coupling is studied experimentally here. 
A: The reason to believe in a Unified Theory is more observational than experimental. Every subatomic particle has mass/momentum and magnetism, while most have charge as well. Obviously those force systems must be cohesive in order to exist. All unified theories seek to explain that cohesion. 
