So, what information did you receive about the decision? You got something, didn’t you? Practically, yes, a Petitioner wouldn’t be likely to appeal an affirmative decision but…everyone should always be made aware of their procedural rights…
Questions: I wouldn’t say that the Sentence is effective in the Tribunal but you can’t licitly act on it unless/until you have the document in hand. That’s a conversation for a faculty lounge, though. It doesn’t really matter as far as getting married. In practice, yes, you will need the document if/when you wish to marry in the Church (see c. 1085.2). It is proof that even though your civil record indicates a prior marriage, the Church regards you as free to marry. That’s the purpose. If you have the document, then you can present it. If you don’t, then the parish could ask for such a document from the Tribunal (I would think).
Dating “licitly” without the document? Well, there is nothing in the law about dating. It seems to me that if you are free to marry, you are free to date…even if you don’t have the ultimate document in hand.
Not sure what I think about the fee structure you mentioned…
Dan