I don't want to be the bearer of bad news, but for quantification, I would stick to quantitative PCR (qPCR or sometimes referred to as real-time PCR). It's called quantitative for a reason.
Traditional/conventional PCR can be used for detection but doesn’t work properly for quantification of the input template. You can quantify the output at the end-point, by agarose electrophoresis for example, but that doesn’t reflect the initial concentration of template.
With qPCR, you’re not interested in the end-point fluorescence which would roughly correspond to the size of the band you’d get with the traditional PCR, I suppose. You are interested in the exponential growth of the fluorescence, proportional to the number of amplicons generated. You quantify the initial input template concentration depending at which cycle the fluorescence crosses a certain threshold (or second derivative). You are not capable of this detection by the classical PCR.
The RNA extraction techniques and reverse transcription reactions are the same for both PCR and qPCR. They are the first two steps in quantifying gene expression, followed by the, I dare to say, gold standard qPCR (until RNA-seq takes over I guess).
If you don’t have a qPCR machine at your disposal in your lab, I would consider asking neighbouring labs to allow you to run your qPCRs there. You can set up reactions by yourself and simply measure it with their machine.
No. Since it doesn't represent the initial copy numbers. The amount of final PCR product does not necessarily have a relation with the initial copy numbers. You can quantify your gene expression data, when you have their Ct values. In addition, you are using paraffin-embedded tissues, which have lost almost all of their RNA content, because of harsh treatments during tissue preparations. These tissues are not suitable even for Realtime PCR techniques. RNA molecule is very sensitive and it has a ultimately low half-life. You should use fresh and well preserved tissues.
It is not possible to use conventional PCR to quantify the mentioned gene. If is possible, use Real-time qPCR and if not, I suggest you measure the gene dosage of STAT3 by considering the type of your samples and the aim of your work because in this way you can use DNA instead of RNA that is easier for you since your samples are not good enough for RNA extraction and in this method you are closer to your hypothesis (although the amount of gene product is more rely on regulation of gene expression!!). To do so, you need to use digital PCR or dPCR that is a biotechnological refinement of conventional PCR that can be used to directly quantify nucleic acids including DNA, cDNA, or RNA. enclosed there is the information related to this method.