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> [...] wouldn't going faster cause you to go higher?

This is exactly what happens, until the thinness of the air reduces your wing lift force and you reach an equilibrium.

> Commercial flights are just perfectly set [...]

There is a lot that goes into this, but commercial planes can change the efficiency of their wing in flight through so-called 'flaps' and 'slats'. These are both a requirement to land at the speed planes land at, or to go faster at lower altitude levels (hence why going faster at lower altitudes is fuel inefficient - it requires you ruining the wing's lift efficiency somewhat. These same speeds are perfectly efficient at high altitudes, because you don't have to use slats/flaps to ruin the wing's optimally efficient shape.

> So I would imagine a fighter jet going several times faster than the speed of a typical commercial flight would then have to nose down to keep from continually rising?

Not a bad thought, but this assumes all wings are equally efficient at producing lift. For an example, the 747 has a lift factor of 17, meaning the 400 ton plane can stay airborne with just 24 tons of thrust. At the far extreme, some gliders have ratios over 70. Fighter plane wings have a typical factor of 2, meaning they literally can only become airborne because of their huge engines.

The wings of fighter jets are actually designed for the speeds they travel at; they use a different shape. Fighter jet wings are typically symmetrical, an extremely inefficient shape (though this also allows them to fly upside down without any loss in lift) which makes it really hard for fighter jets to stay in the air at lower speeds (go look at videos of civilian interceptions to see this in action - the jet's nose will be pointed at the sky).

However, if the fighter jet's wings were of the same asymmetrical design of a jetliners, barring using spoilers/flaps to ruin the wing's shape, they actually would have to nose down to maintain altitude at their increased speed.