Oh boy, this is going to take some time, and some writing. I will need to post it several sections.
Gustav II Adolf (as he is called in Sweden, Gustavus Adolphus was his latin name) did bring several major innovations to the art of war.
Organisation, equipment and recruitment.
Gustav II Adolf's grandfather, Gustav Eriksson (Vasa) created a set of national volunteer units, and Gustav II Adolf built on this system, creating a regiment in each county (or in case of densely populated counties, several, or sparsely populated counties, several counties per regiment). The system was formally voted into law by the estates parliament 1634, two years after the King's death. The regiment was both an administrative unit and a tactical unit - even if only about 10% of the manpower in the Swedish forces in Germany during the 30 years' war were Swedish or Finnish in origin, the regiments created a formal structure into which mercenaries were sorted - to use Swedish organisation and tactics. This was a break with the former system, where mercenaries were recruited as units and created the embryo of a national army. The regiment called up all men between 15 and 50, and free-holding farmers were divided into groups of 10, and each were to supply a soldier - often a second son to a farmer, a crofter or other man that did not own land - initially, most were volunteers, but as the 30 years' war progressed and the population noted how few returned from the war, more and more coercion had to be used to get soldiers for the regiments.
Gustav II Adolf also created a national arms industry, ordering three factories to be set up at Jönköping, Söderhamn and Norrtälje (the latter two got their city rights at the same time). 1622 another factory was established and Örebro, and 1624 one at Norrköping under the leadership of Louis de Geer, who was placed as a leader of all factories 1627. This process was slow and the authorities had problems getting the skilled artisans to move to the factories, regardless of this however, this created an industry that would produce standardised arms, especially muskets and pistols and high-quality gunpowder for the army. The Swedish army could thus often rely on arms that were if not interchangable at least very similar, which streamlined reload times, creating prepared gunpowder charges and thus improved the rate of fire and decreased the amount of misfires.
Swedish cannon production, with the help of artisan immigrants from Wallonia helping Swedish mining and metal industry, became one of the foremost of the world. Swedish cannons were highly regarded - it can be noted that several of the guns from Blackbeard's Queen Anne's Revenge had been cast in Sweden.
Swedish cavalry horses were small (hardly much larger than ponies - 150cm was a common height for a Swedish cavalry horses of the era), ragged, hardy and easy to train to not panic in the smoke, sound and movement of the battlefield. While Swedish noblemen were often ashamed of the low stature and small size of their horses compared to the noble horses European cavalrymen and especially nobles rode, the Swedish horses were much more able to survive periods of illness and low to no fodder than their European counterparts, which resulted in Swedish cavalry often having a lower percentage of dehorsed cavalrymen at any given time in a campaign. Lighter horses might have less weight in the charge, but they often had more stamina and lasted longer on the battlefield before they had to rest.
Rather than focusing on "Why was Sweden so successful in the early modern period?" I would like focus on, "How have Gustavus Adolphus' innovations been perceived over time?" and then answer your two questions in that context.
This is a rather complicated subject, and is intertwined with the ongoing debate on the "military revolution" thesis. Some historians focused on the military reforms in the context of the Thirty Years' War factor the innovations into:
Infantry innovations: greater ratio of shot than pike, faster rate of fire, move toward linear formation, massed volley.
Cavalry innovations: shock tactics, cross-training between cavalry and supporting infantry, move away from caracole.
Artillery innovations: quality, mobility.
Combined arms: mutually supporting units, cross-training, mobile formations that can react to battlefield situations.
Professionalism: large standing army, high quality officers, high quality soldiers, and perhaps less obvious is their seeking to annihilate the enemy in battle.
A lot of these were presented in Michael Roberts' famous article, "The Military Revolution: 1560-1660," published in the 1950s, following his body of work on Gustav II Adolph. This Military Revolution paper focused on infantry and cavalry innovations, in particular the increasing power of shot infantry and the rise of linear tactics, which Roberts credited first to the Maurice Reform in the Dutch army, and then to Gustav II Adolph; and the shock cavalry tactics rather than caracole. The latter further improved the Maurice organization by the aforementioned combined arms optimizations, what with artillery and cavalry.
Geoffrey Parker further argued in "The Military Revolution: Military Innovation and the Rise of the West, 1500-1800" that it was only the infantry's increasing focus on firearms that enabled the military revolution in the early modern era, after all the Swedish were soundly defeated in Nordlingen in 1634 by a veteran, high-quality Spanish army ably led by the Spanish Cardinal-Infante Ferdinand and his cousin HRE Emperor Ferdinand, that was very much still organized and led in the "old Spanish way". Parker argued that it was the rise of professional army that led to the revolution, and this came about due to the appearance of trace italienne fortifications and the need (and ability) for states to fund large standing armies to fight the long sieges.
Going further, it was the Spanish who arguably led this development, what with their Army of Flanders and tercios of Italy. Finally, the Spanish army had also moved more and more towards firearms and away from pikes in organic fashion, which shows in numbers if not in tactical formations and operations. Many top commanders, including Spinola in the early 1600s, had asked for regulation for increasing the number of musketeers in infantry units, only for the Charles III and Charles IV to over-ride their proposals as it was seen to weaken the political influence of Spanish pikemen within the very international Spanish tercio organization. Regardless, the Spanish and Imperial formations had by then moved towards a linear formation, even if they were still centered around pike blocks, although operationally they were very distinct from the "pike blocks with muskets and arquebus in the many mangas or sleeves around the pike blocks" look of the 1560s.
Now, back to your first question: How important was Gustav II Adolph in military history? He is absolutely very important. In addition to leading Sweden's rise in the southern Baltic coast at the cost of the Polish-Lithuanian Commonwealth, he changed the course of the 30YW by turning the tide of Imperial successes prior to the Swedish intervention. Importantly, he brought the war to the southwestern states such as Bavaria, which up to that point had not suffered war on their territory. He was prudent to not attack Vienna immediately after Breitenfeld, instead he wanted to maintain lines of communication and to form an alliance of Protestant states, possibly eventually leading to an attack on Vienna. His early success, death, and subsequent stalemate of Swedish efforts, led to France's direct intervention in the 30YW.
How was he able to accomplish all that? Of course, /u/vonadler/ has written extensively on the state reforms of Sweden, which should not be overlooked. Further, France was contributing significant funds to the Swedish effort. This became important as the original army suffered attrition and they had to hire mercenaries. In turn, the state reforms and investments were originally funded in part using the confiscated wealth of monasteries and churches as Sweden became Lutheran decades prior.
And for your second question: What were his major innovations in the art of war? I won't add further, but rather I'd like to caution once again to first not lose the forest for the trees, and second to truly appreciate not only the organization of the army but rather how it fought in battle. Some will argue about the novelty of the reforms of the former, but it's very hard to argue the impact of Gustav II Adolph's contributions in the latter. Any other army will have capitulated when the Saxons fled the field in Breitenfeld. If they had not, any other army will have been overran when Pappenheim's cavalry flanked the right wing.
So, I hope this post helps put a different perspective, and this post surely leverages the great effort /u/vonadler/ has put into his post.