Why is it that the change in internal energy always uses the formula with Cv in regards to pressure/volume/temperature changes on a gas? Normally I would associate the use of $C_v$ with finding the energy taken into or leaving a system when the volume is kept constant. However, the formula to find $\triangle E_i$ (change in internal energy) is $nC_v \triangle T$. Why $C_v$? Also, does this apply to pretty much anything? Or are there limitations?
 A: I rationalize the use of $C_V$ for finding $\Delta U$ of ideal gases this way: 
Internal energy is the measure of kinetic energy on a microscopic level. That is, the average velocity of all the individual particles that make up a system. 
For ideal systems, the volume of a container which makes up the system will not have any affect on the average kinetic energy of the particles. Think about it, a system of vast volume would still contain the same molecules with the same translational, vibrational, and rotational energies as a system of small size. 
Ideal is important, as if the system were non-ideal, then interactions between molecules via attractive forces and collisions would have different effects depending on the volume of the container. 
For real gases, a vast container would see much more space between particles and therefore less interaction between them. And for a small container, particles will be much closer and the collision rate will be higher. 
Overall, for real gases, we can say it is not true that internal energy  is independent of volume, but the opposite is true for ideal gases. Thus for any ideal gas, $C_V dT=dU$.
A: If you keep the volume constant then the gas can do no  work as $\delta W = P \Delta V = 0$ and so from the first law of thermodynamics the change in internal energy  
$\Delta U = \delta Q - \delta W \Rightarrow \Delta U = \delta Q = n c_v \Delta T$
A: We call $C_v$ the heat capacity at constant volume because that is how it can be measured experimentally, by measuring the amount of heat Q added in a constant volume test and dividing by the temperature change.  But, this physical property that we call $C_v$ has a more general meaning and applicability than that.  In particular, in general, $C_v=(\partial U/\partial T)_V$.  For an ideal gas, U(T,V) is a function U(T) only of T, and not V.  So, the partial derivative becomes a total derivative, and thus, for an ideal gas, we always have $C_v=dU/dT$, irrespective of whether the volume is changing.  But we can still measure Cv directly by measuring the amount of heat Q added in a constant volume test.
A: First, for arbitrary process, this formula applies only to ideal gas. For the special case of constant-volume process, this applies to all gases, by definition of $C_v$.
Now internal energy $U$ is a state function. So if you know temperature in initial and final states for an ideal gas, change in $U$ is thereby completely determined, no matter which process the system went through in between.
