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How big of a volcano would be needed for a large (Alaska-sized) area on an Earth-Like planet to be covered in a thick shell of igneous rock?
Would that volcano also destroy the world or make it uninhabitable?
If this portion of the world was already somewhat flattish, could the igneous crust be pretty unbroken?
The Siberian Traps erupted about 250 million years ago and covered an area of about 2 million km$^2$ with 1 to 4 km$^3$ of rock. That is an area the size of Mexico, or 15% larger than Alaska, and it was enough lava to bury the landscape in something in the range of a km deep of lava.
An alternative type of huge lava event is the igneous intrusion from a rift valley along a separating plate boundary. The most notable of these is the Ethiopian highlands, a giant flood basalt rock thrust up from the Earth. While not as large as the Siberian Traps (about 600,000 km$^2$, or 1/3 the size of Alaska), they are much deeper. The highlands' lowest points are about 1500m above sea level and that is after 30 million years of erosion. The highest peaks are almost 5000 meters. So Ethiopia is an example of a smaller flood basalt that is up to 5 km deep.
No...ish. The Siberian Traps may or may not have been involved in the Permian-Triassic extinction event. The cause of that extinction is unclear, but considering that the largest on-land volcanic event in the last 500 million years happened at the same time is....highly coincidental. Of the three other largest igneous provinces, the Deccan Traps, coincided with the extinction of the dinosaurs, and the Karoo with the Toarchian Turnover, a minor extinction event. So, it seems likely that a volcanic event that big will be bad for life. But, on the plus side, there have been at least four volcanic processes as big as the one you propose, and we're still around?
Probably pretty flat...at first. Since lava flows fluidly, it is going to pour out of the ground and fill all the low points around. So it will be smooth. However, after a few million years, erosion will eat into it pretty severely. The result, after 30 million years for example, would be the Ethiopian Highlands, filled with jagged mountains and deep gorges.
