Is it the gene power the characteristic which influences the frequency of the gene? Because in many cases there doesn't seem to be a connection... so what exactly do gene power and frequency mean?

I believe frequency indicates how likely you are to find the gene in a wild pet. Gene power determines which of the two genes for that slot will be dominant. A gene with gene power 80 will always manifest instead of a gene with gene power 60 or 40. 

@Sumatra_Echo Thanks very much for the help!

Hi,

I have a question related to this: Is the genepower fixed per gene? I adopted a mare who has the brain genes for slow-minded and smart. In the list of genes slow-minded has a power of 80 and smart a power of 90. Yet in her case slow-minded has a power of 90 and it is the dominant gene unfortunately.

How is this possible? 

http://beastkeeper.com/pet/346436

Gene power is usually fixed. However, red genes' power has a chance to increase by 10 if your pet has degeneration. Your mare has Grade 1 degeneration, meaning that she had parents that were related in the last three generations. As for why slow-minded is dominant over smart, even with the same gene power, degeneration also causes recessive bad genes to become dominant.