Gigalaunch and Earth Rotation
Angular momentum is the linear momentum vector ( M * V ) times the perpendicular radius vector R, or ( M * V × R ). The "×" is something called a cross product, which produces an "angular momentum vector perpendicular to the plane formed by the velocity and radius vectors. If you aren't a mathematics or physics adept, you assume everything is at right angles. This is true for a smallish lump of mass orbiting around an axis. The Earth can be considered as a vast formation of whole bunches of lumps of mass, with the centermost mass (ultra high pressure Nickel-Iron) much denser than the crust and ocean and atmosphere above.
You can use calculus and the average density of the earth to compute the total angular momentum of the Earth: 7e33 kg-m²/s. The actual value is lower, because more of the mass is nearer the center than "average". We will use this "optimistic" value to arrive at a gloomy result.
Another "optimistic" assumption is that in the far future, perhaps 500 years from now, interplanetary launch rates will exceed global shipping traffic in 2018, about 12 billion tonnes per year, 1.2e13 kg per year.
This launch rate seems absurd, but imagine asking Columbus whether this much global ocean shipping was possible. The Nina, Pinta, and Santa Maria displaced about 300 tonnes, so this is more than one Columbus expedition per second. His first expedition took 7 months with a crew of 88; 50 man-years. So current shipping rates with 15th century techonology would require at least 1.5 billion crew members, more than 25 times the population of Europe in 1492, 200 tonnes of shipping per European. Half a millenium fron now, a populous solar system may demand far more than this.
Interplanetary launch requires an equator-relative launch velocity of around 11,000 m/s for a Hohmann to Mars. This assumes an eastward equatorial launch, benefitting from about 470 m/s of rotational velocity.
Rocket, mass driver, space elevator, launch loop ... all modalities launching eastward will ultimately have the same effect on angular momentum, transmitting it to the surface of the solid Earth at equatorial radius below. There are complications, but if you follow them all the way to the end, the angular momentum is deposited at equatorial radius, 6.38e6 meters.
So, the "retrograde" angular momentum per year for this prodigeous launch rate is 1.2e13 kg * 1.1e4 m/s × 6.38e6 m, or 8.5e23 kg-m²/s, about 1.2e-10 of the Earth's angular momentum, slowing the rotation ever so slightly, about 4 milliseconds per year. After a millenium of this, a total of 4 seconds per year. The accumulated time error over that millenium is about half an hour. In 7 millenia, the accumulated time error will be an entire day "lost", and the year will be 30 seconds longer. Over "deep time", this may prove a problem.
If we deliver a lot of incoming mass retrograde, that will restore angular momentum. Drag reentry will also heat the atmosphere; the incoming kinetic energy of 1.2e13 kilograms at 11,000 m/s is 7.3e20 Joules per year, or 23 Terawatts. This is a tiny fraction of the 175,000 Terawatts incoming from the Sun, or the heat dissipated by very inefficient rocket launch and fuel manufacture. Solar powered launchloop launch will deposit a small fraction of launch energy (also 23 TW) in the lower atmosphere; the loop is efficient, and if it is powered by stratospheric space solar power receivers, most of the waste heat radiates away from the Earth.
Launching directly towards the asteroids and further out will subtract velocity from the Earth's orbit itself, pushing it slightly closer to the Sun. Launching to a slingshot orbit around Venus will add velocity to the Earth's orbit and subtract from Venus, but opportunities will be rare, once per synodic period, about 580 days. However, the angular momentum of the Earth's orbit is vastly larger than its rotational angular momentum; in mere millenia, no noticable effect.
So ... rotation slowing will affect clocks, but nothing drastic. Earth orbit and atmosphere temperature effects will be very small.