If you have a PDA $A$ that accepts by empty stack you can turn it into an equivalent automaton $B$ using accept/final states. (Some texts explain that both types of PDA are equivalent. Not as you note for DPDA because of the prefix property.)
The new machine $B$ starts from a new initial state and as a first step pushes a new special bottom-of-stack marker $\Box$ under its initial stack symbol $Z$ (by popping $Z$ and pushing $\Box$ and $Z$ back). Now $B$ simulates $A$ using the same instructions and states. Whenever $A$ has empty stack (and is ready to accept, when its input has been read) then $B$ recognizes this because it "sees" the $\Box$ as topmost stack symbol. Then it moves to the newly introduced accept state.
I hope this answers your questions?
The construction of DFAs via (left) derivatives seems not to be as commonly known as it should be; see this paper.
The left derivative of a language $L$ over an alphabet $\Sigma$, with respect to a symbol $a$, is
$$D(L,a) = \{w\in\Sigma^*\mid aw\in L\}.$$
For a regex $r$, let $L$ be the language described by $r$; then $D(r,a)$ is a regex matching $D(L,a)$; one can be computed easily.
The states of your DFA are regexes; a regex $R$ transitions to $D(R,a)$ under symbol $a$. The initial state is the given regex, and the accept states are those regexes which match the empty string $\epsilon$. It's convenient to define derivatives with respect to strings via $D(L,aw)=D(D(L,a),w)$.
For the regex $R=\texttt{/.*(ab|ba).*/}$, we get the following:
$$\begin{array}{c|c}
w& D(R,w)\\
\hline
\epsilon& R\\
a & R|\texttt{/b.*/}\\
b & R|\texttt{/a.*/}\\
ab & \texttt{/.*/}\\
ba & \texttt{/.*/}\\
\end{array}$$
As $D(\texttt{/.*/},s)=\texttt{/.*/}$
for any symbol $s$, no further states are created, so this results in a 4-state DFA for $R$, with one accept state corresponding to the language $\texttt{/.*/}$, the only state whose language contains $\epsilon$. This
agrees with the picture you provided.
Best Answer
Do you know the algorithm for minimizing a DFA? That is what Wikipedia is discussing here.
After minimization, if the resulting DFA has only one state, it either accepts every string, if its one state is an accepting state, or accepts no strings, if its one state is non-accepting.