Yes, proteins like the RNA/DNA binding protein can bind its own messenger RNA and repress their translation. Therefore if they lose this function, for instance by mutation in their functional domain, the protein level is expected to be enhanced.
Yes, some proteins are made ro regulate their own expression, it is called autoregulation. This happens in cyclin related genes. I believe that either p53 or RB is an example.
I think that the question came out wrongly. i meant is there corelatoon between gene and protein expression. ie the corelation between mRna and protein is affected y whatever function the pritein is doing? not the autoregulation.