A theoretical framework for emission originating from rapidly rotating oblate compact objects is described in detail. Using a Hamilton-Jacobi formalism, we show that special relativistic rotational effects such as aberration of angles, Doppler boosting, and time dilatation naturally emerge from the general relativistic treatment of rotating compact objects. We use the Butterworth-Ipser metric expanded up to the second order in rotation and hence include effects of light bending, frame-dragging, and quadrupole deviations on our geodesic calculations. We also give detailed descriptions of the numerical algorithms used and provide an open-source implementation of the numerical framework called BENDER. As an application, we study spectral line profiles (i.e., smearing kernels) from rapidly rotating oblate neutron stars. We find that in this metric description, the second-order quadrupole effects are not strong enough to produce narrow observable features in the spectral energy distribution for almost any physically realistic parameter combination, and hence, actually detecting them is unlikely. The full width at tenth-maximum and full width at half-maximum of the rotation smearing kernels are also reported for all viewing angles. These can then be used to quantitatively estimate the effects of rotational smearing on the observed spectra. We also calculate accurate pulse profiles and observer skymaps of emission from hot spots on rapidly rotating accreting millisecond pulsars. These allow us to quantify the strength of the pulse fractions one expects to observe from typical fast-spinning millisecond pulsars.