“Let's assume that we put the internet ( ) into the elevator and ship it off to space.”
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While total angular momentum is certainly conserved in any rotationally invariant system (read: Earth + space elevator + objects traveling upwards + atmosphere), the situation only for an object traveling to space in the elevator is different: Angular momentum is given by L = I * ω where I is the moment of inertia and ω is the angular velocity. Let's assume that we put the internet ( this video ) into the elevator and ship it off to space. Let's also say that the internet has a mass m and is of negligible diameter compared to Earth's radius. Clearly, its angular velocity ω will stay constant, assuming that the space elevator was built perfectly straight and perpendicular to Earth. However, the internet's moment of inertia I = m r^2 will not stay the same but increase quadratically with its radial position r and its angular moment will thus have to increase, as well. If you take the initial radius to be Earth's radius (~6,300km) and the final radius to be the geostationary orbit (~36,000km), the quadratic dependence will give you an enormous difference in angular momentum which will have to be accounted for by some torque that we apply to the internet. It doesn't matter where this torque comes from--for instance we could imagine the internet to have some kind of jet propulsion engine, where the emitted gas's angular momentum would exactly match the internet's change in angular momentum, so that total angular momentum is conserved. Moreover, it could actually be a small torque, pr