Temperature and nitrogen dependence of 2D carrier mobility in as-grown and annealed Ga1-xInxNyAs1-y/GaAs quantum well (QW) structures (x = 0.32; y = 0, 0.009, and 0.012) are investigated. An analytical model that accounts for the most prominent scattering mechanisms is used to explain the characteristic of temperature dependence of the carrier mobility. An expression for alloy scattering-limited mobility in N-related alloys is developed to explain the behavior of hole mobility for N-containing p-type samples. Analytical modeling of temperature dependence of the electron mobility indicates that N-related alloy scattering and interface roughness scattering are the dominant mechanism at the entire temperature range of interest. The temperature insensitivity of the electron mobility is explained in terms of the overriding effect of N-related alloy scattering and high 2D electron density. A deviation between theoretical and experimental electron mobility at low temperatures is observed not to have any dependency on N concentration. We, therefore, suggest that C-NM interaction parameter of the band anti-crossing (BAC) model must be defined as temperature dependent in order to explain the observed low temperature characteristics of electron mobility. The hole mobility is mainly restricted by interface roughness and alloy scatterings at temperatures lower than 100 K, whilst high temperature hole mobility is drastically affected from optical phonon scattering. Moreover, the hole mobility at high temperatures exhibits an N-independent characteristic and hole density starts to increase at temperatures above 70 K, which is explained using the concept of parallel conduction. Extraction of the hole density in each transport channel (QW and barrier) by using a simple parallel conduction extraction method (SPCEM) shows that, in p-type samples, low temperature hole mobility takes place in quantum well, while as temperature increases barrier channel also contribute to the hole mobility and becomes dominant at high temperatures. The experimental and calculated Hall mobility results reveal that thermal annealing has decreased interface roughness and alloy scatterings.