The coriolis effect is real. Look at the rotation direction of a hurricane in the northern and southern hemispheres, for example, or the direction of "swirl" as water flows down the drain.
The reason why is, the Earth really is a rotating environment; and the acceleration is "fictitious" only from the perspective of an inertial reference frame.
As for "small forces" -- in space, small forces have significant effects because there's no friction to moderate those effects. Thus, solar radiation pressure will be a big deal.
Also, you need to realize that the reason a space elevator would work at all, is because the center of mass of the system is in a geostationary orbit. If you attach something massive to the tether and start cranking it up, the center of mass will be displaced downward, and thus the tether would want to move forward. The response is to crank something upward to keep the center of mass approximately the same; or to fire thrusters to adjust.
Actually, the Coriolis effect doesn't affect anything as small as a drain - it only noticeably affects large-scale phenomena such as hurricanes (as you also point out). In your sink, the direction of swirl is determined by the design of the fixture, and it's entirely possible to have drains in your house that swirl in both directions.
Two words: Ballast Weight.
Increase it slightly so the centre of mass is always above geostationary orbit position. Tension on the cable is slightly greater, but it's always stable.