That's pretty much correct. "Finally" can mean the same as "at last", "ultimately", "in the long run". It implies waiting a long amount of time before something happens so that when it does, it has "finally" come. Usually, however, "finally" would refer to the very last thing that happens.

The Term: Finally Used like that it is an exclamation. Example: Finally, I have finished my exams. Finally! I can start my homework. It suggests that you have been occupied with something before you could reach this goal. In these examples, the goals were either to finish the exam or to finish a previous task before you could start your homework. You could say here: "Finally, I have the answer I have been searching for!" But of course it can also be used as you have described "I have finally completed my homework!" or "I have finally finished my exams!" but the previous word order with "finally at the beginning of the sentence (Finally, ... ) suggests a greater sense of relief.

No, I think it is correct, I'm listen this words always in players.