I don't know that I would draw that kind of distinction. There were plenty of conflicts in Europe alone that officers could draw from to develop doctrine. Russia fought against the Ottomans in the 1820s, the Poles in the 1830s, and had a host of Central Asian and Caucasian wars throughout the pre-Crimean War period. The British and French dealt with innumerable overseas conflicts - their "small wars and big riots" - to the extent that the French military, at least, made a conscious adoption of anti-doctrinal doctrine, that is to say: little to no prescribed response to a given set of circumstances, but a focus on experience and contextualization to solve problems, because of the belief that the environments in which the French army had to fight were so varied that it would be impossible to construct a set of rules - or even rules of thumb - to follow in all circumstances. (Christ, that's a long sentence.)
One major thing about the nineteenth century - not just the period between 1815 and 1853, but the whole time - was the increasing emptiness of the battlefield. Engagements like Waterloo or Leipzig had been fought at relatively close quarters for pretty much the entire time. Close-order infantry formations dominated the battlefield. But as the century went on and weapons technology improved to the point that rifled firearms could reliably hit targets further and further away (and a few associated technological developments, like the Minié ball), armies began to spread their troops further and further apart. Close-order formations got rarer and rarer. This was an evolutionary shift, not an on-off switch. And it's most visible in the 1850s and 1860s, because those decades featured a lot more war - or at least more-publicized war - than those on either side.
Other alterations after Napoleon are more context-sensitive. For instance, some armies, like the American military, raised engineering and fortification to an art form during that period, influenced by officers like Dennis Hart Mahan. This was not, at least initially, a response to improved killing power of battlefield weapons, but a synthesis of lessons supposedly learned from the last set of wars - Mahan drew on the lines of Torres Vedras, the Russian fortifications at Borodino, and the American fortification of the Baltimore harbor for examples. His disciples, like George McClellan, supplemented that with personal experience of the value of fortification in the Crimea. The Americans employed battlefield fortification - and siege - at considerable length in the Mexican War, and would do so again in the Civil War. But the Americans - and to a lesser extent the French - were somewhat unique in that regard. Prussia, for instance, spent very little attention to fixed fortifications, and only developed an interest in battlefield fortification during the 1870 war. In Austria, engineer officers seized on fortification as a less pricey - and therefore easier to get past parliamentary scrutiny - alternative to military expansion or improved training. The German Confederation as a whole refused to spend any more than the legally required amount on the fortifications at Mainz and Luxemburg.
About the second question: procurement. Depends on the country, obviously. Earlier in the period, new inventions were usually marketed directly to the military by manufacturers - wouldn't make much sense otherwise. Later on, armies started to put out contracts and seek bids for near-future technology. In 1870-71, the German military carried out trials to determine the replacement for the M1842 needle gun, with the primary competitors being the Bavarian Werder M1869 and the Mauser Brothers M1870, with Mauser eventually being chosen (everywhere except Bavaria). The German government eventually developed a very close relationship with the Mauser firm, as you might have heard.
In addition, technology was more critically evaluated than you might expect. Many armies were unwilling to screw around with untried tech or gear that might not stand up under battlefield conditions. After the 1859 war, the Austrians converted all of their artillery over to four- and eight-pounder rifled bronze cannon, specifically eschewing the cast-steel processes that Krupp was pioneering for the Prussians because of concerns with the steel guns' performance. Austria also rejected breech-loading artillery because of similar concerns with the ability of manufacturers to make an airtight seal. These concerns were legitimate - in 1866, many Prussians carrying model 1842 Dreyse needle guns would be injured by failed seals on their rifles - and by no means indicated that Austrian procurement believed those technologies would never be viable - just not right now.
