does it matter if we model a deterministic or non-deterministic automat

Any exercise will state whether you should build a deterministic or non-deterministic automaton. If not, please ask and we will clarify this in the exam on the blackboard.

Is ending in a wrong state (not a final state) equivalent to reading letter σi in state si where no si+1 for this letter exists?

No that are two different things: If a deterministic automaton is not in a final state after reading an input sequence, it simply means, that the input was "not accepted". That is a valid computation, since the automaton's job is to tell us whether an input sequence is accepted or not (is part of the described language or not). That is different from having no successor state. If there is no successor state (e.g. you read *'a'*, but there is no outgoing transition with *'a'*) then this is not a valid computation.

In deterministic automata this can't happen, since we must have exactly one outgoing transition for each symbol, for each state (take a look at the transition function \delta).

In non-deterministic automata this is ok, since the transition function \delta maps to a set of possible successors (which can be the empty set), and acceptance is defined via "exists a path ... " rather than one path.