Target localization based on frequency diverse array (FDA) radar has lately garnered significant research interest. A linear frequency offset (FO) across FDA antennas yields a range-angle dependent beampattern that allows for joint estimation of range and direction-of-arrival (DoA). Prior works on FDA largely focus on the one-dimensional linear array to estimate only azimuth angle and range while ignoring the elevation and Doppler velocity. However, in many applications, the latter two parameters are also essential for target localization. Further, there is also an interest in radar systems that employ fewer measurements in temporal, Doppler, or spatial signal domains. We address these multiple challenges by proposing a co-prime L-shaped FDA, wherein co-prime FOs are applied across the elements of L-shaped co-prime array and each element transmits at a non-uniform co-prime pulse repetition interval (C$^3$ or C-Cube). This co-pulsing FDA yields significantly large degrees-of-freedom (DoFs) for target localization in the range-azimuth-elevation-Doppler domain while also reducing the time-on-target and transmit spectral usage. By exploiting these DoFs, we develop C-Cube auto-pairing (CCing) algorithm, in which all the parameters are ipso facto paired during a joint estimation. We show that C-Cube FDA requires at least $2\sqrt{Q+1}-1$ antenna elements and $2\sqrt{Q+1}-1$ pulses to guarantee perfect recovery of $Q$ targets as against $Q+1$ elements and $Q+1$ pulses required by both L-shaped uniform linear array and L-shaped linear FO FDA with uniform pulsing. We derive Cramér-Rao bounds (CRBs) for joint angle-range-Doppler estimation errors for C-Cube FDA and provide the conditions under which the CRBs exist. Numerical experiments with our CCing algorithm show great performance improvements in parameter recovery.