I understand it was Napoleon's suggestion that Karl I be elected Romania's first king. Who 'elected' him, how was Napoleon in a position to offer this suggestion, and was this practice of 'parachuting' foreign nobles in to rule countries not of their birth common at that time?
A bit of background.
If you try to look at the lists of Moldavian and Wallachian princes from the Middle Ages towards 1859, you'll be struck by their diversity - lots of ruling people, lots of ruling families, no system of smooth transition until the Ottomans took charge and began imposing their own candidates. So when the nationalist intellectuals from the early 1800s started to dream about an united modern nation that would replace the old principalities, they had a couple of things on their mind: a desire to make a nation similar to modern western states (like France or Belgium) and the conviction that local elite rivalries would destroy the delicate system of governance that would be set up for the purposes of modernizing the region. Give the throne to the Ghyka family, the Sturdzas would be its enemy; give the throne to the Sturdzas, you'd have the Cantacuzino clan to deal with. Seems far-fetched? Well, you'd only have to go to neighbouring Serbia in order to verify the validity of the position - 2 local clans (Obrenovic and Karagheorghevic) battling out for supremacy.
The solution to elect a foreign prince was first expressed in 1848, during the revolutionary period, but its most important moment came after the Crimean War when the Great Powers invited the two Principalities to form local assemblies in order to vote on their future (the so-called "adunari ad-hoc"). Among their chief requests, the one concerning a foreign prince, from the ruling families of Europe, held a pretty significant place.
The same assemblies also asked for an united nation. This was unacceptable for many reasons, two of them called the Ottoman Empire and Austria. One was the official sovereign of the region, one held a romanian-majority region (Transylvania). So the conclusions of the assemblies were ignored and elections were held for both principalities. Which promptly led to the election of a single guy in both - Alexandru Ioan Cuza. Not quite in the spirit of the wishes of the Great Powers but without technically breaking the directives.
Alexandru Ioan Cuza reigned for 7 years. During his time, he made significant reforms, managed to create several unified institutions for the two states he officially ruled but his main adversaries based their opposition towards the "Domnitor" on the fact that they had once voted in favor of a foreign ruler. Incidentally, Cuza had voted for the same thing and even during his reign he reinforced his belief in the need for a western-born prince.
Cuza was ousted through a coup in 1866. The conspirators were from both conservative right and liberal left ("monstrous coalition") with disagreements in every imaginable field - economics, social reforms, political institutions. But, as it turns out, they were all fans of foreign leadership. Their first step was to offer the Romanian throne to Philippe, Count of Flanders (future father of King Albert I) - who promptly refused. Meanwhile the 7 Great Powers were gathering in Paris, deciding against any foreign Romanian Prince and even putting in jeopardy the future of an united Romanian nation.
The situation was dire. But Karl I was really the sole solution - he was supported by Bismark and King Wilhelm of Prussia but also had the approval of Napoleon III. Even so, the other Great Powers were commited to not recognising him. His arrival in Romania was done in secrecy, avoiding Austrian police and for the rest of the year Romanian diplomacy had to deal with the ramifications of its unilateral decision (spolier alert - it managed to eventually convince the European Powers to recognise Karl's I legitimacy on the Romanian throne).
For your last question - the practice of "parachuting" foreign nobles was a common European practice since the Middle Ages. The current dynasties of Spain, Great Britain or Belgium can be counted as examples of this policy. If you're speaking only about the Balkan region, you have only four examples - Greece (1832), Romania (1866), Bulgaria (1879) and Albania (1914).
Related question: why were German princes often tapped to rule new countries (like Romania) or countries seeking a new monarch (like Spain), instead of naming a native ruler?
You may be interested in my old answers relating to this phenomenon more broadly:
How do royal families become royal in the first place? a short answer which covers some of the Balkans.
a discussion of how various Balkan and Middle Eastern royals families got chosen from the outside between the mid-19th century and the end of World War I.
a discussion of how this happened at the end of World War I but did not happen at the end of World War II, for instance with a restoration of a Korean royal house, when the geopolitics were different.
In short, it was quite the norm during much of the “Long Nineteenth Century” for foreign nobles not just to parachute into countries where they weren’t born, but to be specifically brought in by great powers for that purpose. The exceptions (Zog in Albania, for instance) generally had to defeat a rival prince selected by the great powers. But in many countries, this was accepted, as to a certain extent, having a royal family meant having a certain legitimacy. This norm fell apart rather quickly after WWI and was not repeated after WWII.