Relation between enthalpy and Cv
For an ideal gas, your equation for $\Delta U$ applies not only to constant volume processes, but to all processes.  It's just that, in experiments to measure Cv, it is very convenient to use a constant volume path because, under those circumstances, no work is done, in which case $\Delta U$ is just equal to the amount of heat added.  So this gives you a direct measurement of Cv.

As far as $\Delta H$ is concerned, the correct equation is $\Delta H=\Delta U+\Delta (PV)$.  But, from the ideal gas law, $\Delta (PV)=nR\Delta T$.  So what does that give you for $\Delta H$?