Genoa was one of several Italian maritime republics of this period (the best known being Venice). They used their very powerful navies to get rich in the Mediterranean, both because they were able to protect their merchants from piracy and because the larger European powers wanted to be friendly with them to have access to that naval power. Since they relied on sea power, their expansions focussed on grabbing useful ports rather than big swathes of land. They would often make agreements with the larger powers to set up a colony on the coast in their territory, which suited the larger power in question because it meant the republic’s ships would be bringing valuable goods to and from that port.

Theodoro is a bit of a historical oddity — it was the last Gothic state. As in the Germanic people that burned Rome down way back at the fall of the Western Empire, roughly a thousand years before this map. When they migrated into central and western Europe, some of them went to Ukraine and eventually into Crimea instead.

Crimea remained part of the Eastern Roman Empire up until the Fourth Crusade, in which France and Germany were supposedly going to go capture Jerusalem. The entire Fourth Crusade was a colossal shitshow which resulted in the crusaders partitioning the ERE instead, and the Crimean Goths basically just bounced between whoever happened to be the main power in the region for the next few hundred years. They managed to more or less maintain independence so long as they paid tribute to whoever was in charge that year (first one of the ERE’s successor states, then the Mongols, then a Mongol successor state, then a successor to that) until they were eventually conquered by the Ottomans a few decades after this map.

if i am not mistaken, the genoese colonies in the black sea were a mix of conquest and purchase following the fourth crusade and the consequent fall of eastern roman power in the region, with the purpose of dominating trade, going from the black sea to the aegean islands (such as mytilene, on the map) onwards to liguria

as for theodoro, it was a monarchical city state that popped up from the break up of the golden horde trying to put it in a simple manner, after the fourth crusade, theodoro exchanged hands a couple of times, but every time the hold over it was weaker and weaker, which led to the city mantaining eastern roman law and practices and, eventually, independence, kind of making it an eastern roman rump state theodoro was also notable for its large gothic population, which survived as a distinct ethnicity up until roughly the 18th century

pretty sure tana is not modern azov city, as google maps shows the ruins of the former several kilometers north of azov

I dunno, ask a British person.

I only know about them thanks to Age of Empires II