Would non-GM crops such as Grains, Fruits, & Vegitables retain their non-GM make up if they were to be re-planted in soils that have been made entirely of composted GM plants and GM plant matter? Or would the GM plant matter still retain those properties enough mutate the transplanted or planted crop?
The process of genetically modifying crops is performed in a lab. Genes would not leak into planted organic crops from compost any more than they would move from decomposing soil nematodes, earthworms, or manure. So no, organic plants would not be genetically modified. However, the labeling of a crop as "organic" is not always standardized, so it could be a cause of concern for some people who do not understand the process.
This is called horizontal gene transfer. In general, it's quite difficult to get a plant or animal to pick up foreign DNA and integrate it into its own genome, requiring special reagents and laboratory conditions. That's why we didn't have GMO fifty years ago. Note that you will spend your entire life consuming plants and animals for food, but you won't integrate cow, apple, or potato DNA into your own genome.
However, some viruses and bacteria can and do swap DNA with each other, and with their host organisms. In particular viruses can carry DNA into eukaryotic cells, and can even integrate the DNA into the host genome. Specially engineered bacteria and viruses are typically used as the vectors to create GMO in the first place. There has been concern about horizontal gene transfer from GMO organisms, but it's widely thought that it would be rare enough to not be a significant hazard. As far as I know there in nothing about GMO that would make them more susceptible to horizontal gene transfer in your compost pile than non-GMO.