How can we know what degree of polynomials to use in a certain situation?
We use mostly cubic ones to join 2 points because they give us enough freedom for a smooth joining.
Going too high leads to Runge Phenomenon (worsening because of too much oscillation of the curve)
Yes, it seems that few people use higher than cubic
Right---one typically sticks with low-order polynomials in order to avoid oscillations, but there are polynomials specifically designed to avoid Runge's phenomenon, like the Chebyshev polynomials.