Very well, because you asked. More musings with little actual calculation or actual engineering experience to back it up.
So we have a hollowed out shell, 3000 km in diameter, made of soft, frangible material. But let’s pretend that moon rock is strong enough to withstand the stress, or that our alien beings have thoughtfully reinforced it with alien Duct Tape so that we don’t have to worry about falling scree.
What do we structurally reinforce our sphere with? 2x4’s aren’t going to cut it. Neither will steel. Maybe carbon nano-fibers or Ringworld * scrith * would do the job. We’re still talking about a Civil Engineering nightmare. Gravity wants to make it fall down. Tides want to make it disintegrate. And now you have to worry about pressure variations as well.
Do we keep a vacuum in the center of your hollowed out sphere? So much of it is going to be out of the atmosphere as it is. You have to cope with the pressure of the atmosphere on the bottom 100 miles or so, but that’s minor compared to the other forces involved. I’m pretty sure you don’t want to pressurize the interior to help support the structure – for one, we’d run low on atmosphere. For another, you’d probably have to compress it to well beyond the tensile strength of the structure. And finally, you’d hate for it to spring a leak.
The most obvious choice is to put in a few several-kilometer wide vent holes and let the pressure equalize – but there are some problems with this approach.
As the rotation of the sun takes the moon shell into and out of sunlight, it’s going to alternately become very hot or very cold. (It may in time equalize. Dunno.) Add thermal stresses to your list of civil engineering problems. The air contained in the shell is going to undergo pressure changes as the temperature of the shell varies. Depending on the temperature equilibrium achieved, we might see gales of scorching hot or bitterly cold winds emerging from the vent holes. If we vent the ocean waters as well, as suggested in the OP, those winds would be either raging thunderstorms or blizzards.
What else does the shell do to the Earth’s weather? Messes it up good, I will think, even if the moon doesn’t act as a giant heat sink and either freeze or burn us all. Off the top of my head, we’ve got a 3,000 kilometer mountain blocking ocean and wind currents. At the very least, we have severe and unpredictable climactic changes and can stop worrying about piddling changes due to Freon and greenhouse gases. The huge volume of the sphere is going to block sunlight from much of the surrounding ocean. Vast areas will see sunlight only 1/2 the day. Some areas will be pretty much in perpeptual eclipse. Without solar radiation, the equatorial regions will drop in temperature. Another arctic region will form, surrounded by a ring of perpeptual hurricanes. (It’s not going to be easy to get to this tourist mecca.)
But what about the reflected sunlight hitting the moon’s surface? Some of is going to reflect back on Earth. Maybe instead of polar regions we’ll have extreme temperature variations? Or both? Packing for a trip to the Pacific is going to become very challenging.
How about electromagnetic effects? I’m going out a bit on a limb here, but I’m thinking that if you have an enormous structure, one end in space, and one not, surrounded by howling winds, you might see a few billion EV of static build up. The sphere might be surrounded by a ring of constant lightning. Very pretty and it would save on the lighting bill. Just don’t plan on ever using broadcast transmissions in that hemisphere ever again.
I may have missed a few aspects of the impossibly strong superstructure scenario, but the most positive thing you can say about it is that it doesn’t obviously spell the end of all life on earth as we know it. It just makes it far, far more interesting…