No state entity controls Step stones as of now. I am at work so can't find references right now, will do when I am free.
As you noted, Stepstones used to be the land bridge which connected Essos and Westeros. It is the very land bridge which the First Men used during their great Migration. So it stands to reason that along the way, some First Men clans may have chosen to settle right there instead of taking the roads further inside Westeros. We already know that Westeros at that point was devoid of any humans and was populated solely by Giants and Children of the Forest.
With the First Men advancing rapidly and exterminating settlements of the Children, they decided to strike back by breaking the land bridge to stem the flood of new comers using their brand of Water magic. But it was too little and too late. There were already enough First Men inside to wage war and the bridge itself wasn't broken completely, it just turned into a series of isles between the two continents, aptly named as Stepstones.
Importance of the Isles
The location of the Isles gave them strategic importance in the Naval "Great Game" of Planetos. The Isles sit right on the sealanes from Narrow Sea to Summer Sea, controlling all the trade flowing from West to East and vice versa. Which is why the Isles changed hands a number of times.
Valyrian Freehold and her Daughters
Valyrian Freehold, the first example, didn't show much interest in the Stepstones as they already had a naval base to control the said sealanes in the neighbouring Island of Tyrosh. After the fall of the Freehold, her daughters fought over the possession of the strategic Islands because whoever controlled them, practically controlled the trade flowing from East to West and Vice versa. Given that the Free Cities are mostly merchant cities, it was particularly important for them to gain control of the Islands. Neighboring Myr and Lys clashed a number of times for the Isles and they changed hands rapidly.
Haaar! There be Pirates!
Capitalising over exhaustion of the Free Cities due to inter-war and century of blood, Bands of Pirates moved in and established their control on the Isles and started making riches by looting the trading ships.
Of course then Essos was plagued with ambitions of Volantis to become new Valyria and the Essosi were too busy to pay attention to the Isles any more. And the Westerosi mostly stayed out of that part of the world. And then by the end of that era, Westeros was conquered by Aegon the Conqueror.
Before the conquest, Given the nuance caused by the pirates and due to importance of the Isles, the neighbouring Westerosi realms from Stormlands and Dorne also engaged in warfare over the Isles but like the Free Cities, they too failed to make their gains permanent and Isles once again became hub of Piracy.
After the conquest, As raids continued from the Isles on Westerosi land and shipping, Prince Maegor (Later King Maegor) retaliated by a punitive campaign against the pirates. But he did not annex the Isles into the domain of Iron Throne.
When the Free Cities got done with Volantis, three of them, Lys, Myr and Tyrosh united to form the Triarchy to contain Volantis in future. They looked West and invaded the Isles to end piracy. Westerosi nobles were generally in support of the actions of the Triarchy.
Kingdom of Stepstones and the Narrow Sea
Losing all hopes of inheriting the Iron Throne due to birth of three sons and two daughters to his elder brother King Viserys II, Prince Daemon Targaryen, with support of Westerosi Lords now disgruntled by heavy tolls imposed by the Triarchy, invaded the Isles with his band of adventurers, who like him were seeking lands and riches. Daemon crowned himself King of the Stepstones and the Narrow Sea. But now Dorne got involved because they were absolutely not going to let the Targaryens forge another Kingdom right across their shores. The tables turned when Dorne and Triarchy joined forces. Daemon, sensing the inevitable doom of his Kingdom while there were interesting prospects rising in Westeros which culminated into Dance of the Dragons, abandoned his Kingdom. But the Kingdom lived on with his followers taking the place. Five men ruled in that capacity until Dorne and Triarchy completely obliterated the Kingdom.
When Triarchy collapsed, another pirate claimed the title of King of the Stepstones during reign of Aegon III after the Dance of the Dragons.
But eventually, the Kingdom collapsed once again with Pirate Lords ruling their own independent Islands. This state continues to this day however in light of the recent events, there are reports that another man has claimed the title of the King of the Step Stones. Rumors are that he might be the fugitive Admiral Aurane Waters who deserted King Tommen's navy with his newly made dromonds.
So in conclusion:
- No region has any de-jure claim on the Islands as of now except perhaps Lys and Myr but they are not interested in pressing their claims any longer.
- The Isles are independent.
- The population of the Isles seems to be a mix of Rhoynar (Different from their Dornish Kins as they did not join Nymeria on her way to Dorne), Westerosi and Essosi settlers, mainly pirates.