Can a machine choose whether or not to accept a state in a FSM
The diagram appears to be of a Nondeterministic finite automaton, or NFA. A NFA is a model of a theoretical machine that accepts a given string if any path through the NFA for that string can end up at an accepting node.

So the simple answer to the question in the title is "yes, if you're talking about NFAs". (As opposed to Deterministic finite automata, or DFAs, which must have a unique outgoing edge for each character at a given node)

One traditional way to write a program to check if a string is accepted by an NFA is to keep track of all the states that the machine could be in after seeing a particular character, and then at the end declare that the string is accepted if one of the states we could be in now is an accepting state.

For example, consider the input string abbbbb.

At the start, the machine can only be in state $S_0$, so the set of states it can be in is $\{S_0\}$.

After a, the machine can now only be in $S_1$, so $\{S_1\}$.

After ab, the machine could be in $S_1$ or it could be in $S_3$, so $\{S_1, S_3\}$.

After abb, the machine's set of possible states is $\{S_1, S_2, S_3\}$.

After abbb, the machine's set of possible states is $\{S_1, S_2, S_3\}$.

After abbbb, the machine's set of possible states is $\{S_1, S_2, S_3\}$. You should see a pattern by now.

And in fact, at the end of the whole input string abbbbb, the machine's set of possible states is $\{S_1, S_2, S_3\}$, and that contains an accepting state ($S_3$), so the string is accepted.

The regular expression should now be easily readable from the machine: ab*(a|b)