You are correct: the Earth is not an inertial reference frame. It is an accelerating reference frame, and that makes any projectile calculations very complicated. The good news is that, for almost all practical considerations, treating it as an inertial reference frame is more than close enough.
Of course, it is a tenet of physics that all of the laws of physics apply equally to ANY inertial reference frame. There is no absolute motion, so any linear motion can be considered to be a valid frame of reference.
Of course, it is a tenet of physics that all of the laws of physics apply equally to ANY inertial reference frame. There is no absolute motion, so any linear motion can be considered to be a valid frame of reference.
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They do indeed take it into account, but only for projectiles that go very far (think ICBM's and spacecraft). There is correction due to the earth's orbit from the Coriolis effect, just as there is from earth's rotation. But since its proportional to the angular velocity of the motion, it's about 365 times smaller.
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If you wanted to be completely exact (disregarding quantum mechanics) you would have to take in not only the Earth's motion but also the total graviational force applied by everything in the universe as gravity has infinite bounds, albeit very miniscule at large distances but if you take into account the trillions of other stars in the universe or even the stars in the milky way it would alter a calculation to some extent. It is just so small that it is almost never "needed" to prove a formula or experiment, i.e. it lies far within the deviation of the measuring devices.