John R. Elting’s Swords Around a Throne is the book on the Grande Armée, to my mind; I’ll be drawing on it a lot in this answer. It covers the French Army from the pre-Revolutionary Royal Army to just after Waterloo, although the focus is on the years from 1802-1814 (there’s a separate section dealing with the Hundred Days). It covers each topic sequentially, from the contributions of the Royal and Revolutionary armies, to logistics, staff-work and intelligence, all the way through to weapons and battlefield tactics. I highly recommend it.

As for your specific questions:

• Did the Grand Armee have a better system for training recruits than other armies?

As with all questions about the Grande Armée, this comes down to the period you’re looking at. The army of 1805 was considerably different to the army of 1809, 1812, and 1813. In the first campaigns, troops were extremely well-trained. Most battalions had a fairly high proportion of veterans, and most general officers had served through the Revolutionary wars. Even so, Napoleon instituted a complete training regimen in the camps and Boulogne, going right back to basics of individual and company drill and working up to movements at the Corps d’Armée level. In this way, the entire army got to train in its assigned form of warfare, and the fact that all the soldiers were in one place aided the cross-pollination of ideas that had been one of the strengths of the Revolutionary armies – tactics developed on the Rhine or in the Alps would transfer with the commanders that had served in those places, and might find new uses elsewhere. Training and drill was constant: when troops weren’t marching or fighting, they drilled – even in the winter of 1806/07.

By 1809, however, things were different. The original Grande Armée had been badly battered by years of war, and greatly reduced by the 1807 campaigns in Poland. Further, most of the best troops in the French Army were in the Iberian Peninsula when Austria struck, meaning that the troops levied hurriedly by Napoleon were mostly green. ‘Cadres’ (a selection of officers and NCOs, preferably well-behaved veterans, to instruct new conscripts) were drawn from depot battalions and the Imperial Guard – this practice was increased as the demands for fresh units increased. By 1813, faced with war with almost all of Europe, the army taught its recruits the bare essentials – forming square, forming column, and shooting – before throwing them into the fire to learn everything else. The Marie-Louises of 1814 often didn’t even get that.

• Why was Napoleon able to move faster than other generals? I am aware of how his Battalion system facilitated the movement of soldiers, but I don't know how he transported equipment and supplies.

At a basic level, French soldiers generally marched farther/faster than other European soldiers did. The average day’s march for French troops was about 15 miles; for Austrians, ten miles. Further, the French logistical systems were intentionally less extensive than other armies – that is, where other armies would have long trains for food and other supplies, the Grande Armée would in general only carry with it its ammunition, especially when on forced marches – rations would be issued prior to the beginning of a manoeuvre, and units would hope to capture enemy depots and thus eat the enemy’s food. Further, French troops were experts at foraging and living off the land: this ability grew out of the early Revolutionary Wars, where the Republic was near-incapable of feeding its armies (Napoleon’s first assault into Piedmont in 1796 was as much motivated by the desire to stop his army from starving as it was to knock Piedmont out of the war).

This meant that the French Army did not need a long supply train stretching back to their homeland, as Wellington had to organise in the Peninsula. This meant that French troops could move faster and that they were capable of operating out of their lines of supply for short periods, allowing them to move into an opponent’s rear. Thus Napoleon’s army was perfectly-calibrated for his preferred method of war – a lightning strike to the enemy’s rear to capture his depots and disrupt his communications, with further fast marches cutting off all avenues of escape and forcing the enemy to either give battle on unfamiliar ground or capitulate.

This changed in Russia, with an extensive supply system being necessary (although French troops were still able to forage in Russia before the retreat). However, this method was never Napoleon’s preferred method of resupply.

• Did the Grand Armee use better equipment than other armies?

Not particularly; its musket was almost identical to enemy muskets and it frequently used enemy ammunition; its guns were rather lighter than enemy guns, but it made extensive use of captured guns, which rather negated that advantage; swords and other hand weapons had their own advantages and disadvantages, but they were far less important than differences in morale, leadership and training. The quality of its horses varied over its lifetime, and it never really had enough of them – but this did not diminish the effectiveness of the Cavalry, since attempts were always made to capture enemy horses.

• Was the morale of his troops really higher than that of his enemies?

Morale is a tricky thing to quantify, and again it varies by period; the troops of 1805 and 1806 were largely veterans, led by veterans and blooded under the Emperor. As the Napoleonic Wars continued, more and more fresh drafts of conscripts were added and morale could become a problem. The practice of cardreing fresh regiments with drafts from the Imperial Guard and other seasoned units helped, as did Napoleon’s reputation: the Duke of Wellington famously said that Napoleon’s presence on the field was worth 40,000 men; his reputation as one of the most formidable generals alive was, by 1806, well established. While he might lean on Merovingian, Roman, and Charlemannic imagery in his stately role as France’s monarch, to his soldiers Napoleon was a general; he was addressed as Mon Empéreur, as if the Imperial title was a military rank. He was an expert at man-management, and expected his senior generals to cultivate this skill: occasionally capable of remembering the faces of individual faces years after their first meeting, given to impromptu promotions or monetary rewards to particularly brave men. Napoleon was a soldier to his bones, and his army respected this above all other qualities.

Further, the French Army had a tradition of victory. Not only had it defended the Republic through the Revolutionary Wars, it had thrice humiliated Austria under Napoleon’s command; in 1806 it had smashed the army built by Frederick the Great in 33 days; it had humbled the warriors of Holy Russia and fought them to a standstill. French troops had an extremely high espirit de corps, but their morale was not invincible: as Napoleon himself said:

The French soldier is the most difficult of all to lead. He is not a machine to set in motion, but a reasoning being which you must govern… The French soldier loves to argue because he is intelligent. He is a severe judge of his officers’ ability and courage… When he approves of the operations and respects his commanders there is nothing he cannot do… The French soldier is the only one in Europe who can fight on an empty stomach [but] he is more demanding than any other when he is not in combat… A French soldier has more interest in winning a battle than a Russian officer… the art of retreating is more difficult with the French… A lost battle destroys his confidence in his leaders and incites him to insubordination… [He is] humiliated… The French soldiers only motivation is honour; it is that motivation which must be the source of punishments and rewards.

Further measures were taken to stiffen morale. The Légion d’honneur offered both monetary reward and all-important honour, as did presentation weapons and the individual distinctions of arm and unit – grenadiers, the elite storming-troops, had a flaming grenade on their cap badge, while the light infantry prised their red epaulette. All units prised their eagles, and would fight to the death to defend them. The possibility of promotion also encouraged men: a sous-lieutenant Marcel rose from conscript to officer in a mere four years, and his case was not exceptional.

In addition, many of the factors that motivated the Revolutionary armies persisted into the Grande Armée: while France might no longer be a Republic, its soldiers were still free citizens of La Patrie and fought as such. Foreign troops were more variable, with the Swiss and Poles generally being courageous (and, in the case of the Swiss, extremely disciplined), Italians and Germans variable, and Spanish and Dutch generally (though not exclusively) less fire-eating than the French.

While none of these factors might add up to necessarily quantifiably higher morale than that of its enemies, it did produce an army that, when its Emperor demanded, could produce incredible results.

Sources:

Swords Around a Throne: Napoleon’s Grande Armée, by John R. Elting

The Campaigns of Napoleon by David Chandler

Napoleon’s Greatest Adversaries: The Archduke Charles and the Austrian Army, by Gunther E. Rothenerg

Redcoat: The British Soldier in the Age of Horse and Musket, by Richard Holmes

Rifles: Six years with Wellington’s legendary sharpshooters, by Mark Urban

(as a note: the linked thread is locked, so you wouldn’t be able to answer the question. Sorry.)