Can a deterministic language be accepted by a deterministic Push Down Automaton?
Sure, a deterministic language can trivially be accepted by a nondeterministic PDA. Simplest way to see it is that a deterministic PDA is just a special case of the non-deterministic one. If that isn't enough, for some transition

$\begin{align}
   \delta(q, a, A) = \{(p, \alpha)\}
\end{align}$

you can add the following, without changing the language accepted
(the modified PDA isn't deterministic, it is enabled to do completely pointless $\epsilon$ moves):

$\begin{align}
   \delta(q, \epsilon, A) = \{(q, A)\}
\end{align}$