Forecasting noisy data with polynomials
Starting with your last question: to extrapolate you do just use your fitted coefficients and extend them to a greater $x$ range.

However, I think this approach is inherently flawed for data such as yours over any significant time period, especially when using a polynomial model. The problem is that any polynomial has a limited number of turning points. Once you pass these the data will tend to head off to $\pm \infty$ which is unlikely to model anything well.

Frequency domain analysis may be a possibility, but I expect you would run into similar problems. Another approach could be to use some sort of moving average or auto-regressive model.

Banks and similar financial institutions invest millions into modelling this sort of data so it definitely isn't easy.