What is the frequency distribution of each base in a DNA sequence?
The frequency of the bases in the genome isn't equal to 0.25, the frequency depends on what kind of organism you mean. However let's take a look at some of them:


bacteria, most of the time we can see a bias towards some bases, this could be a GC bias, for example if the bacteria lives in extreme conditions, because GC can vorm three hydrogen bonds compared to AT which can only form two hydrogen bonds. So it's genome is more resistent to extreme temperature, less denaturation chance because the stronger bonds. In bacteria there are also some polymerases which have a bias towards bases, I won't discuss this in depth to keep it short, but from these two points we can already conclude the genome isn't random
Homo Sapien, for example we need CpG Island near our promotor region, to regulate transcription. Further we have higher GC% in gene coding regions to reduce mutation chance, because we don't want to mutate genes wich are needed for survival. Their are also some sequence which are very specific for example TATA box and Poly-A signal, if our genome would be random, so all base would have a frequency of 0.25 these specific sequence could occur at place where we don't want them or they wouldn't occur at places where we want them. We need a promotor (with TATA box) for RNA polymerase binding, if our genome was random we have a big chance we would have genes wihtout a TATA box so no transcription of these genes will occur


There are a lot  more properties of the genome we could discuss but I think this will give you enough insight why the bases in the genome don't have a equally alike frequence of 0.25

UPDATE:

in addition to your second question: I think the chance of finding a genome which has an equally alike distribution is very small. because almost all organisms have undergo evelution, and when we have evolution there is natural selection and natural selection relies on mutations. So (almost) all genomes have udergo mutation, however this sometimes can favor an equally distibution for example if our genome is AGAA and one A mutates to an G we will have an equally alike proportion. So to summarize: maybe you can take a look at NCBI and look for a gc 50% wich means 50% AT so 25% of each base. If you find a genome wich has exact 50% gc, you have found your answer