The reason why the specific heat at constant volume is used even though it is not a constant volume process is because it happens to be that change in internal energy for an ideal gas, for ANY process, is given by
$$\Delta U = C_v\Delta T$$
This can be derived as follows:
For a constant pressure process:
$$\Delta U=Q-W$$
$$\Delta U=C_p\Delta T – P\Delta V$$
For one mole of an ideal gas
$$P\Delta V=R\Delta T$$
Therefore
$$ \Delta U=C_p\Delta T – R\Delta T$$
Also, for any ideal gas
$$R=C_p-C_v$$
Substituting
$$ \Delta U=C_p\Delta T – (C_p-C_v)\Delta T$$
$$\Delta U=C_v\Delta T$$
Concerning the equation for R, $C_p-C_v=R$ for one mole of an ideal gas (n=1)
We can prove that $C_p-C_v=R$ is by using the basic definitions of the specific heats and enthalpy, combined with the ideal gas law.
Specific heat definitions, ideal gas (they are actually partial derivatives holding P and V constant, respectively):
$$C_p = \frac {dH}{dT}$$
$$C_v = \frac {dU}{dT}$$
Definition of enthalpy (H)
$$H = U + PV$$
For one mole of an ideal gas, ideal gas law
$$PV=RT$$
Therefore
$$H = U+RT$$
Taking the derivative of the last equation with respect to temperature:
$$\frac {dH}{dT} =\frac {dU}{dT}+R$$
Substituting the specific heat definitions into the last equation, we get
$$C_p – C_v = R$$
Hope this helps.