Understanding decidability
1) The automaton "halts" after it reads the input, whether or not it reaches a final state.  It accepts the input if, when it halts, it's in a final state.

2) Inputs are always finite strings.  There are generalizations of finite automata ($\omega$-automata of various kinds  [Buchi, Rabin, Muller]) that accept $\omega$-words (countably infinite strings) as inputs.