Davout will be one of the few people I could ever speak of as a historical figure with honest respect. The Marshalate is full of shady and disreputable characters but Davout holds his own as not only one of the best Marshals in France but a uniquely capable commander that always made the impossible, possible.

This is the man, that Napoleon had, a uniquely capable and skilled commander that did the impossible. This is the man Napoleon feared most. The Napoleonic mythos always portrays the Emperor as a dynamic, almost romantic hero that came forth to push the cobwebs of the Ancien Regime but in reality, he could be as petty and fearful as any Monarch. As such, Napoleon was always fearful of anyone that he thought might or had potential to depose by assassination or military coup as there were multiple attempts on his life and at least one semi successful coup in winter of 1812. One reason Napoleon didn't allow or teach his Marshals the basics of Independent Command (something Davout was naturally gifted in) was a fear of the French army no longer needing him and therefore replacing him, something that is minorly echoed in the writings of David G. Chandler, a well known Napoleonic scholar.

With this, where does Davout stand? On numerous times Davout literally does the impossible to save Napoleon;a straight 36 hour march to Austerlitz, fighting half the Prussian army with only a third of their numbers in one corps at Auerstadt, being an important anchor in almost every major battle during the 1809 campaign, and as you mention the masterful defense of Hamburg. Time and again, Davout was shown to be a master at commanding his men.

However, one problem appears, Napoleon the blamer. Napoleon rarely took credit for his mistakes and few ever lived through the Napoleonic Era with a positive note to their name from him (only those that died in battle, such as Lannes or Bessiers, escaped with a positive mention with the exception of some like Suchet, whom is one of the others I can say lived without great sin in the era), while others like Ney get complains about being "too immoral and too dumb to lead more than ten thousand men." Davout also was blamed for "making mistakes at the Moskova", but was blamed more for the Emperors inactivity than any actual reason.

With all this, you can see that Napoleon, the vainglorious and jealous human that he was, was more than likely jealous and fearful of Davout, one of the few people who honestly would have followed Napoleon to the literal gates of hell. It does more credit to Davout than Napoleon that Davout was loyal to the end, literally requring an envoy from Louis XVIII to stop the siege of Hamburg.

Napoleon may not have trusted Davout, but Davout wholeheartedly trusted Napoleon.

Sources:

Napoleons Marshals ed. By David G. Chandler

The Campaigns of Napoleon by David G. Chandler

Napoleon: A Life by Andrew Roberts.

Edited for corrections

Part I

This question is one of the more contentious topics in Napoleonic military historiography and given the Emperor's defeats in 1813 and 1815, it tempting to wonder if Davout had been in the place of either Ney, Oudinot, Grouchy or Soult, things might have gone differently for Napoleon. But as with many counterfactuals involving an individual's presence or absence, such a fictional scenario assumes that these replacement figures would have triumphed over the many obstacles facing their historical counterparts. It is certainly not outside the realm of possibility that Davout would have been defeated outside of Berlin for the same reasons Ney and Oudinot failed (an overstretched French army, poor cavalry, a resurgent Prussian army, Napoleon's preoccupation with capturing Berlin, etc,) . As with all counterfactuals, what could have happened is and will remain unknown. But in the case of Davout in this period, what is known is that Napoleon tasked him to defend Hamburg and the northern German coast and placed him as Minister of War during the Hundred Days, and in these tasks, Davout was successful.

Looking at Davout's employment through the prism of the 1813 and 1815 defeats tends to obscure the fact that Davout significantly aided French strategic concerns where he was. The 1813 campaign was a case in point. In February 1813, as rumors of the French defeat in Russia became confirmed, popular anti-French revolts gripped Hamburg and French power in the city seemed to be on the verge of collapse throughout the month. The approach of a small force of Cossacks led to an open rebellion among the city's inhabitants and an evacuation of the French troops under Carra St. Cyr to Bremen.

Napoleon was livid over the surrender of the city both for short-term and long-term strategic reasons. The Hanseatic cities had been annexed to France in 1810 and the fear for Napoleon was that as one major city went, the rest of the Northern European ports would likewise fall. This would have meant Napoleon's rear flank would be open to intervention and the Russians and Prussians would have ready access by the sea to Britain and Sweden. The subsequent landing of 500 British troops at Cuxhaven seemed to confirm this fear of an insecure rear area. From the perspective of Napoleon's long-term goals, Napoleon saw the Hanseatic coast as an integral part of his empire and French control of the continent's commerce. Thus there both immediate and long-term reasons, Napoleon needed a strong arm in Hamburg to restore order. Davout was in many respects ideal for this task. Not only was he a marshal with experience at independent command, but he was well aware of the northern German coast after being stationed there in 1810 to facilitate the region's formal absorption into the French Empire. In a letter to his stepson Eugene, Napoleon outlined his strategic rationale behind this decision:

My son, I am sending you a newsletter from Hamburg dated the 12th: you will see that 200 Cossacks are going to seize the entire 32nd Military Division [a military administrative unit consisting of the three French departments: Bouches-del’Elbe, Bouches-du-Weser, and Ems-Superieur- with Hamburg as its headquarters]. They cost me millions; they spread the insurrection to all that is behind the line of the Elbe. This is the result of you standing on the defensive behind the left bank; you no longer have any effect on the enemy. I have repeated this; I hope you finally feel the need to occupy a camp before Magdeburg and threaten to cut off all enemy units that attempt to infiltrate from the side of Hamburg, at eighty leagues from the main body of their army. This is even more important because what occurs in the 32nd [Military] Division is likely to attract landings by the English, Swedish, and perhaps even the Russians; and, if the enemy establishes himself on the lower Elbe, you will have nothing to oppose him when he makes a movement in strength against Dresden and you will find yourself without a line of operation.

Davout accomplished the task of both reoccupying and pacifying the city and while keeping it from falling under Allied control. Davout managed to keep a good deal of the northern German coast under French control until French military fortunes imploded in the autumn of 1813. Although Davout managed to hold the fortress for France until the Restoration in 1814, it should also be stressed that he did so at the expense of the city's civilian population and his governance of the city was exceptionally brutal, even by the standards of the day.

Just as stationing Davout in Hamburg suited Napoleon's strategic intentions in 1813, and there is evidence that Napoleon envisioned a much more active role for Davout after the Battle of Dresden, there were similiar imperatives at work placing Davout as Minister of War during the Hundred Days. Although Napoleon presented his return as a triumph wholeheartedly embraced by the French populace, the reality was more complex. A good many of the marshallate did not rally to their former master and both the southern and the western regions of France were fairly Loyalist. Moreover, the loyalty of those who rallied to Napoleon was questionable and highly conditional. For example, the famous Revolutionary Minister of War Carnot rallied to the Emperor, but also proposed that Napoleon make concessions to liberal politics and eventually accept constitutional limits on the Emperor's power. As in 1813, Napoleon needed a loyal individual in his rear area while he was in command of the main French army. When Davout traveled to the Tuileries for an audience with Napoleon, the Marshal requested a military command, but the Emperor informed Davout that he was needed in Paris. According to one report, Napoleon confessed to Davout despite the proclamations of Austria's solidarity with Napoleon, "The truth is that this is false, that I am alone, alone before Europe. This is my situation! Will you abandon me?"

Davout agreed to be Minister of War despite his reluctance and in this position he helped to stabilize Napoleon's rear and raise and equip a large army in a short time. As in Hamburg, Davout's knowledge of these matters and skills as an administrator made him the natural candidate for this position. Moreover, Davout had demonstrated consistent political loyalty to Napoleon, something Napoleon appreciated after his first abdication when a Parisian camarilla centered around Talleyrand managed to depose the Emperor for his violations of the French constitution. To Davout's credit, he managed to accomplish a pacification of France and ensured that the Armée du Nord and Armée de la Réserve were well-equipped in a short time frame. It is telling that of those French elites that initially rallied to Napoleon, Davout was the only prominent member of the government of the Hundred Days to argue for continued resistance after Waterloo.

Yet there were other factors at work behind Davout's isolated commands in these years. Although Davout's reputation grew postwar as France's undefeated marshal, his military skill was less clear from the perspective of 1813. Davout's I Corps did not exactly cover itself with glory in the Russian campaign and came out of the retreat incredibly battered. Additionally, Napoleon became annoyed with Davout's purported desire to become the King of Poland after the Russian campaign. According to the correspondence collected by Marshal's daughter, Napoleon rebuked Davout, saying:

Davout, gossip has it that you have become obsessed with ambition and that you are working to become king of that country.

Other testimonials of 1812 also speak to a growing distrust between Davout and Napoleon over the former's egoism and desire to be elevated to a king. These suspicions also exacerbated the factionalism between the marshals. Both Ney and Berthier scapegoated Davout for mistakes made during the advance and retreat into Russia. This factionalism colored Napoleon's evolving relationship with Davout, which was a mercurial mixture of praising the Iron Marshal's fortitude, but also rebuking him of mistakes, both real and imagined. Davout's frigid and somewhat acerbic personality also did not also endear him to his fellow marshals and he tended not to work well with them. This was particularly important in the Hundred Days when many of the marshals and flag officers that rallied to Napoleon did not much care for Davout. Ney still bore a grudge against Davout for his purported abandonment of the III Corps during the retreat from Russia and Rapp's dislike of Davout stemmed back to 1810/11 when Rapp critiqued Davout's treatment of the Hanseatic ports when he was in command of the Army of Germany. Soult also held himself as Davout's equal and there was a minor row in early May 1815 when both men presumed to issue orders to the Armée du Nord before the Emperor took command. Although Napoleon clarified the command structure, Davout wrote back to Soult on 22 May with a rebuke of Soult's presumption of command:

I ask you sir, to abide by the decision of the Emperor. In general, if you give orders on your part and I on mine, the results can only lead to the greatest confusion, and i will resign my position as Minister of War. I accepted the ministry reluctantly, and it would give me the greatest pleasure to remove from myself this burden.