Is a PDA's stack size linear bounded in input size? I was thinking as follows: At each step, a PDA can put arbitrary many symbols onto the stack. But this number is constant for every individual PDA, so it can't be more than, say, $k$ symbols per step. Using only regular transitions, the stack can rise to maximally (more or less) $kn$ symbols in a run on an input sized $n$.

But what about $\epsilon$-transitions? Is there a simple argument why their maximum number should as well be independent of the input size?

So, in short: Is a PDA's stack size linear in the input size?
No. In NPDAs, you can have cycles of $\varepsilon$-transitions that add symbols to the stack. Thus the stack content can be unbounded.

Proving that CFL ⊆ CSL via automata is tough; the trusted route via grammars seems advised.