Request for examples regarding PAC learnability
You can find a proof for the equivalence between PAC and agnostic PAC learning in theorem 6.7 of "Understanding Machine Learning" by Shai Ben-David and Shai Shalev-Shwartz. The proof goes through showing:


$\mathcal{H}$ is PAC learnable $\Rightarrow$ $\mathcal{H}$ has finite VC dimension (lower bound on sample complexity).
$\mathcal{H}$ has finite VC dimension $\Rightarrow$ $\mathcal{H}$ is agnostic PAC learnable (upper bound on sample complexity)


If you don't care about the bounds being sharp, $1$ is not too hard, but $2$ is definitely beyond the scope of this site. Combining the above results with the fact that agnostic PAC learnability implies PAC learnability concludes the proof.

In regard to your second question, let $l_1$ be the 0-1 loss function and $l_2$ be a loss function which is nonzero only on a finite subset of $\mathcal{X}$ and is the 0-1 loss in that region (suppose you only care about very specific mistakes). Take any class with infinite VC dimension, then it is not agnostic PAC learnable with respect to $l_1$, but is learnable with respect to $l_2$ (If the loss function is nonzero only on a finite region, you can treat this case like having a finite hypothesis class).