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Why is oxygen placed on top of hydrogen? Does this have anything to do with letting the warmer and not the colder liquid have a non-spherical bulkhead and thus more surface area to reduce boil-off?

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Having the colder tank lower in the stack is marginally helpful for the thermal design, but in the case of the space shuttle, aerodynamic considerations dominate.

For aerodynamic stability, it's desirable to put the center of gravity well forward of the center of aerodynamic pressure (which means roughly "where the drag is happening"). Think about a throwing dart, with its heavy forward end and draggy fins at the back; the drag force on the fins keeps it from tumbling.

The wings of the space shuttle orbiter are a big source of drag, so the center of gravity needs to be forward of that. Liquid oxygen is about 16 times as dense as liquid hydrogen, so forward it goes, to move the overall CG forward of the wings.

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    $\begingroup$ It also reduces the range that the engines have to gimbal $\endgroup$
    – Hans
    Commented May 13, 2019 at 13:52

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