The current idea is that all life evolved from a single cell. Analysis indicates the probability of two different origins being very unlikely, given shared DNA. However, why couldn't numerous organisms have evolved and then conjugated, with a few lineages surviving? It would vastly increase the odds of modern organisms sharing the DNA that they do.
Because they would not use the same DNA codons. Codons are extremely conserved across life, with only minor variations, there is no reason this should be the case unless those codons are inherited from a common ancestor. This is contingent on what you mean by organism, prior to the incorporation of ribosomes it would be more possible.
If what you suggest had occurred it would drastically DECREASE the odds modern organism would share the same DNA.