Translation:The father wants to give the spoon to the girl.
After να, we use the aorist subjunctive (looks like the simple future) to talk about one-time things, and the present subjunctive (looks like the present tense) to talk about continuous or repeated or habitual things.
Here, the father wants to give the spoon as a one-time event, so we use the aorist subjunctive να δώσει.
θέλει να δίνει would be something like "he wants to give (regularly)" or "he wants to be giving (over a long time)".
So in this sentence στο κορίτσι would be accusative? What would το κουτάλι be?
το κορίτσι and το κουτάλι are both in the accusative case.
το κορίτσι has the preposition σε "to" before it, which merges with the article το into στο.
Since you're learning German, you may be familiar with such "giving" sentences having the thing given in the accusative case and the recipient in the dative case.
Ancient Greek had a dative case, but lost it (except in some fossilised expressions). It's mostly replaced by the preposition σε + accusative case -- a bit like how in English, the recipient is often expressed by "to" + objective case, as in "I will give the book to him".
You are technically right, but they sometimes call that thingy that follows the auxiliary verb in the perfects by the infinitive (only for modern Greek). έχω δώσει - you can find in some grammars that this δώσει is called an infinitive; but δώσει in the phrase αυτός θέλει να δώσει is simply a subjunctive (an aorist subjunctive).