If I use a regularization (e.g. L2) can I not apply early stopping? I've seen that early stopping is a form of regularization that limits the movement of the parameters $\theta$ in a similar way that L2 Regularization penalizes the movement of $\theta$ to be closer to the origin.
Does that mean that I can avoid overfitting if I use a regularization other than Early Stopping and train for more than the epochs that it takes to overfit the model as prohibited by Early Stopping?

Why not? You can use several methods of regularization at the same time, for example Elastic net combines $\ell_1$ and $\ell_2$ regularization. Different regularization methods work differently, so combining them sometimes can be lead to better results than using any one of them alone (like with Elastic net).
Notice that those regularizations work differently. With $\ell_2$ you essentially assume a Gaussian prior for the parameters that push them towards small values. Another example of regularization is dropout which forces the model to "do the same without those parameters". With early stopping, we train the model as usual but stop the training when we see a drop in validation set performance. With the two previous examples of regularization, you explicitly chose how you want to influence the model (make weights smaller, force building smaller "submodels") while with early stopping, your model "doesn't know" that you want to regularize it anyhow. It is perfectly fine to combine them because they do completely different things. We call them all regularization, but they are far from doing the same thing.
Additionally, early stopping makes the training faster since you potentially skip some iterations, which could be an added benefit in a scenario where your model trains for a long time. In some cases, this alone would be a reason to use early stopping nonetheless if you use other regularization.
