I saw an Alaska-flaired mod here and have an Alaska-flaired question here - a match made in heaven. Also I'm trying to get my alumni library card form printed and it just won't load, so I'm asking a question in hopes that my university-website-rage subsides.
Basically, question is as reads. Were there major Russian settlements in Alaska? Are they still there? Was Alaska run by a governor appointed by the Tsar?
Hey there. Is it me you're looking for?
In any event, I'm happy to help. The quintessential book on Russian America is Lydia Black's Russians in Alaska: 1732-1867, and this answer draws on that book as well as a few others. I'll address your questions in order.
Russian America was governed via an odd blend of corporatism, imperialism and anarchy. It's important to remember that Russian America covers almost a century and a half of history, and its history changed just as much as that of Europe and other settled polities.
From the time of its "discovery" by Europeans to 1799, the situation was chaotic, with various trading firms exercising authority with little if any oversight by the imperial government. Power came (very literally) from the barrel of a gun, and whichever trading firm had the guns at any given point (chronologically and physically) had the authority.
In 1799, the Russian-American Company (itself a product of a merger between trading companies) was granted an imperial monopoly over Russian America and the valuable fur trade that everyone was after. For the 88 remaining years of Russian rule, the RAC and its shareholders held tenuous authority over portions of coastal Alaska.
From 1790, when he arrived to command the Shelikov-Golikov Company (one of the ancestors of the RAC) to his death in 1818, Aleksandr Andreevich Baranov was the de facto governor of Russian America. As Black says (and I agree), "If anyone among the empire builders may be considered the creator of Russian America, it is Aleksandr Andreevich Baranov ..."
It was Baranov who shepherded the RAC through five hungry years in which no Russian ships arrived in the colony. It was Baranov who led the RAC's recovery from a series of devastating battlefield defeats at the hands of the Tlingit. It was Baranov who wrested profits (at times violently and brutally) from Native and foreign competitors in the fur trade.
Baranov's capital, such as it was, was New Archangel, today's Sitka, Alaska. In the first years of the 19th century, the Russians had been granted a trading post near the site of modern Sitka by the resident Tlingit. The Russians made themselves so unwelcome that in 1802, an alliance of Tlingit clans attacked and completely destroyed the trading post. Most of the Russians and their Native hunters (from the Aleutians and Kodiak archipelago farther west) were killed.
Fortunately for Baranov, the defeat coincided with the arrival of the first Russian expedition to circumnavigate the globe. Reinforced with European arms and soldiers, the Russians and Baranov counterattacked, defeating the Tlingit at Sitka and turning the site of the victory into their capital in Russian America.
The victory was somewhat of a hollow one, as for the next several decades, the Tlingit executed an effective blockade of the island on which Sitka is located. Russian fur hunters and their clients had to ride aboard foreign vessels (usually American) to pursue fur hunting and trading as far afield as Baja California. By the 1830s, the valued fur otters had been all but hunted out, and only this economic factor relaxed the blockade as the Tlingit were unable any longer to resist Russian attempts to penetrate Southeast Alaska.
In the meantime, things were changing in Russian America. The RAC operated on a series of 20-year charters ─ its monopoly was not permanent ─ and Baranov's death in 1818 coincided with the end of the first charter. Baranov was replaced by a man named Hagemeister, a capable naval officer who was a military man first and a businessman second. After Baranov, every subsequent Russian governor was a naval officer. Their duty was first to the state and second to the economic success of the RAC.
At no time up to 1867 were there significant numbers of Russians in Russian America. The RAC intended for Alaska to make it money, and every Russian in the territory was someone they had to pay to support (and this included priests and the soldiers stationed in the area). Instead, much of the labor was supplied by the Creole class, a half-Native, half-Russian group that was prized in Russian America but discriminated against in both Russia and America. New Archangel was a tiny town but the largest in Russian America. Second-largest was the town now known as Kodiak, and third-largest was the town now known as Unalaska.