This sentence is in my summary of a short story:
"Luke luckily picks ducks at a game-booth and wins Chuck's wife Margo a carnival prize before he dies shortly after in a car crash."
Here is the too long, didn't read version -
tl;dr - It's correct, but if I were to rewrite it, I would phrase it like this:
Luke wins Chuck's wife Margo a prize by luckily picking ducks while playing a carnival game; shortly afterward he dies in a tragic car crash.
The long version:
Let's break it down into parts of speech and see if it holds up grammatically when broken down into it's individual parts.
Luke picks ducks and wins prize, (for) Margo, dies.
This is the basis of the sentence. "Luke" is the subject. "Picks", "wins", and "dies" are the predicate verbs. "Ducks" and "prize" are the direct objects, meaning the verbs they receive action from are transitive. "Dies" is intransitive and has no object. "Margo" is the indirect object that answers the question "for whom" regarding the prize. So far so good.
"Luckily" is an adverb modifying picks and logically modifies it. "At a game booth" is an adverbial prepositional phrase describing where the action took place, thereby modifying "picks", "at" is the preposition, "a" the article for "game booth" a compound noun in this instance and object of the preposition.
On to the next part of the compound predicate. "Wins" being the second verb Luke does. "Prize" is the direct object of "wins" here. "Carnival" is an adjective modifying 'prize." "Margo" as mentioned earlier is the indirect object for the second verb in the sentence "wins" and the ultimate receiver of the direct object "prize." "Chuck's" is strange thing here, a noun acting as an adverb, qualifying "wife." Still, it logically works within proper syntax. "Wife" again is a strange thing, but a common usage as a noun acting as an adjective qualifying "Margo." All that said, the second part of the predicate is logically and syntactically sound.
Finally, the third part of the compound predicate. The intransitive verb "dies" here is encased in a prepositional phrase. Let's break it down. "Before he dies shortly after..." is an adverbial prepositional phrase here further modifying the verb "wins" as it is currently places in the sentence.
Here is where the sentence construction becomes a bit unclear. First, though, I'll break down the phrase.
"Before" is the preposition. "He" is the object of the preposition, and a pronoun properly connected to the antecedent "Luke." "Dies" is an intransitive verb here but also a subject complement describing Luke's state of being through the use of the pronoun "he" as the connector. "Shortly after" is an idiomatic expression in this sentence acting as an adverbial phrase modifying "dies." Notice it has a preposition inside it "after" that lacks an object. If one were being technical, this would be considered an adpositional phrase, either circumpositional or postpositional. Finally "In a car crash" is an adverbial prepositional phrase, modifying "dies." "In" is the preposition, "a" the article for the object "crash," and "car" is the noun acting as adjective qualifying "crash." It can be argued that "car crash" may be classified as a compound noun.
So there you have it. Syntactically the sentence is a proper construction using prescriptive rules. It is an awkward sentence in terms of style, but it's not incorrect. It could be more concise.