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How do we go up and increase lateral speed without bending the cable into a big "C" shape?

That's the biggest issue.

The concept behind the space elevator is that the string is acually moving in synch with the earth's rotation, which means you have orbital velocity at geostationary altitude (22,300 miles). Most of the designs I've seen go up to about 36,000 miles, at which point the terminus is at earth escape velocity (in order to keep up with rotation). Letting go of something at that point would fling it off into interplanetary space. As you pointed out, though, at 100-200 miles altitude, the string is moving at much less than orbital velocity and anything released there would fall back to earth quite quickly.

However, anything rising up the cable would need to be accelerated to keep up with the string - and that would slow the string, eventually leading to collapse. Something would need to be done to replace that angular momentum - every time the 'elevator' went up. That would take as much momentum transfer (fuel) as it would take to orbit the same amount of mass using conventional means, unless you could recover and de-orbit as much as you're lifting into orbit. Right now, there's no concept on how to do this.

It's also not trivial to produce the required strength. Actually, carbon nano-tubes are about an order of magnitude less than the required strength-to-weight, and they're nano-tubes. Just a bit short for the job. Carbon-carbon bonds, as used in the nano-tubes, as also the strongest bonds known within current chemistry/physics. Getting that extra order of magnitude is not a job for engineers, it's a job for basic science - a breakthrough in theory, not just a refinement of technology.

Nonetheless, I believe both problems can be solved. I'm not sure the economics will ever work out, though. Like a lot of problems, it's more social than technical.

they're nano-tubes. Just a bit short for the job.

They're also getting longer. Was just a few years ago that "buckyballs" were first created/discovered, and didn't take very long before they were extended into "nanotubes" a few millimeters long. Just a matter of time before they're cranked out by the mile.

carbon nano-tubes are about an order of magnitude less than the required strength-to-weight

Strictly speaking, there is no minimum required strength-to-weight ratio. In theory, the cable could be built out of anything so long as it is tapered enough (i.e. the point where it meets the earth has to be strong enough to hold the amount of payload to be transported up the cable; higher points need to be progressively wider to carry the weight of the payload plus the weight of the cable up to that height). The catch is that the taper ratio is a function of the strength-to-weight ratio. Carbon nanofibers, if they can be mass-produced, brings the taper ratio down to the point of being merely difficult (something on the order of 100:1, IIRC) rather than downright prohibitive.