Note that DNA is NOT a self-replicator. It requires ribozymes (RNA + Proteins) to actually replicate. To my knowledge there has not yet been a enzyme made of DNA (thermodynamically it's favoured structure is not catalytic).
to my knowledge this is what happens when DNA is replicated:
they take the tangled dna string thats inside the core of a cell, it is transported outside of that core and then some enzyme copies the DNA but it replaces the T protein with a U (could be another one). This is then called RNA.
Why do they do this? well the T protein is very vulnerable. By replacing it, they can now safely transport the rna to the place where they're duplicated.
What you are trying to describe is transcription, the process of reading a DNA template to produce a RNA transcript. This is done in the nucleus for eukaryotes, and in the nucleoid for prokaryotes (which don't have a nucleus). RNA Polymerase is the protein that synthesizes mRNA (from the DNA template) and the nitrogenous base Uracil (U) is used in place of Thymine (neither are proteins).
In fact Uracil is less stable that Thymine and the hypothesis is that DNA uses Thymine due to its lower rate of point mutation. (Note I mean less stable in the sense that it can be mutated more easily due to functional groups).
Citation needed.