The electron injection process into a plasma-based laser wakefield accelerator can be influenced by modifying the parameters of the driver pulse. We present an experimental study on the combined effect of the laser pulse duration, pulse shape, and frequency chirp on the electron injection and acceleration process and the associated radiation emission for two different gas types—a 97.5% He and 2.5% N2 mixture and pure He. In general, the shortest pulse duration with minimal frequency chirp produced the highest energy electrons and the most charge. Pulses on the positive chirp side sustained electron injection and produced higher charge, but lower peak energy electrons, compared with negatively chirped pulses. A similar trend was observed for the radiant energy. The relationship between the radiant energy and the electron charge remained linear over a threefold change in the electron density and was independent of the drive pulse characteristics. X-ray spectra showed that ionization injection of electrons into the wakefield generally produced more photons than self-injection for all pulse durations/frequency chirp and had less of a spread in the number of photons around the peak x-ray energy.