We report a Raman spectroscopy study of graphene field-effect transistors with a controlled amount of defects introduced in graphene by exposure to electron-beam irradiation. Raman spectra are taken at T=8K over a range of back gate voltages (Vg) for various irradiation dosages (Re). We study effects in the Raman spectra due to Vg-induced doping and artificially created disorder at various Re. With moderate disorder (irradiation), the Raman G peak with respect to the graphene carrier density (nFE) exhibits a minimum in peak frequency and a maximum in peak width near the charge-neutral point (CNP). These trends are similar to those seen in previous works on pristine graphene and have been attributed to a reduction of electron-phonon coupling strength (D) and removal of the Kohn anomaly as the Fermi level moves away from the CNP. We also observe a maximum in I2D/IG and weak maximum in ID/IG near the CNP. All the observed dependences of Raman parameters on nFE weaken at stronger disorder (higher Re), implying that disorder causes a reduction of D as well. Our findings are valuable for understanding Raman spectra and electron-phonon physics in doped and disordered graphene.