Why did David execute the man who brought him Saul's crown?

Amalekites were cursed by God (Ex 17:14). God commanded Saul to exterminate them but he didn't. Samuel rebuked him for disobeying (1S 15:18).

Later, the Philistines invaded. Saul was badly wounded and probably killed himself.

2S 1:

1 After the death of Saul, when David had returned from striking down the Amalekites, David remained two days in Ziklag.

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David continued to carry out God's command to exterminate the Amalekites.

Ironically, an Amalekite [A1] came to David and claimed to have finished Saul off at his request. He told David this story:

8 "And [Saul] said to me, ‘Who are you?’ I answered him, ‘I am an Amalekite.’ 9 And he said to me, ‘Stand beside me and kill me, for anguish has seized me, and yet my life still lingers.’ 10 "So I stood beside him and killed him, because I was sure that he could not live after he had fallen. And I took the crown that was on his head and the armlet that was on his arm, and I have brought them here to my lord.”

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According to A1, Saul knew A1 was an Amalekite. Now David knew also. A1 was trying to gain favor with David.

13 David said to the young man who told him, “Where do you come from?” And he answered, “I am the son of a sojourner, an Amalekite.”

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A1 affirmed that he was an Amalekite.

14 David said to him, “How is it you were not afraid to put out your hand to destroy the Lord’s anointed?” 15 Then David called one of the young men and said, “Go, execute him.” And he struck him down so that he died. 16And David said to him, “Your blood be on your head, for your own mouth has testified against you, saying, ‘I have killed the Lord’s anointed.’”

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Why did David execute A1 who brought him Saul's crown?

David didn't kill A1 for bringing the crown.

Was it because David thought that this man should have gotten Saul some help?

No, Saul's bodily condition was beyond help.

Was it because he was just angry and sad and human and did something rash because he lost his friend Jonathan?

No. David would not have transferred this anger onto A1. He wouldn't just take it out on A1. David was cool.

The only question he asks the man after he finds out that Saul and Jonathan are dead is "why were you not afraid to strike the Lords anointed"? So I wonder is it related to him thinking that this man is disrespecting God?

Right, something like that. Ostensibly, it was because A1 killed the Lord's anointed. Saul was the Lord's anointed. A1 didn't respect the Lord's anointed. By executing the man, David demonstrated that such an act against the Lord's anointed would not be tolerated. David himself had refrained from killing Saul twice (1S 24, 26). David, too, was the Lord's anointed. This execution would deter people from killing David as well.

Privately, David did not believe A1's motivations were pure. A1 might have lied to him. Furthermore, Amalekites were cursed by God.

Right after the execution,

17 David lamented with this lamentation over Saul and Jonathan his son, 18 and he said it should be taught to the people of Judah; behold, it is written in the Book of Jashar. He said:
19“Your glory, O Israel, is slain on your high places! How the mighty have fallen!

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David's feelings for Saul and Jonathan were genuine. A1 didn't have this kind of respect for Saul. So David executed A1, an Amalekite.