There's a basic nearly fatal flaw in your workflow. Many of the lines in a computer display are 1 pixel wide, a lot of the text is composed of single pixel wide fonts and the standard DV frame size of 720 X 480 will not work well with single pixel wide lines in any form. You're also scaling down the screen which reduces the thickness of the lines so the quality drops.
I'd use a screen capture program that allows you to resize the output. On the Mac my favorite is Screen Flow, on the PC it's Camtasia. Do most of your editing in those apps. Screen flow does a much better job of scaling down the screen than Final Cut.
If you have access to Premiere Pro you could try that for scaling down. It uses the GPU for scaling instead of the pixel interpolation that FCP uses so the quality is better.
You can find Screen Flow with a quick google search.
Well, you are working with a wrong comp size, so there will always be different scaling ratios in both dimensions horizontally and vertically. There is simply no way to cleanly fit a 16:10 ratio in a 16:9 comp. It will always look odd. Aside from that, there is no point in vague statements like "the quality drastically drops". Doesn't mean anything to anyone around here and aside from your wrong size you could work with wrong framerates, make a mess of fields, use bad compression settings or what have you. Could just as well be OpenGL issues with hardware accelerated previews in both programs. So provide screenshots and clarify...
I used Screenflow to record the screen, and exported it in full quality, 1680x1050. Then animated it with Photoshop layers. I realized now, if I make a new comp using the NTSC preset of 720x480 and drop my already animated video that I exported from AE(, the1680x1050), and leave it untouched, (where it's scaled up too much), then the quality is great, but when I resize it, using the 'V' tool, I loose quality, as you can see below...
Before I scaled to fit
After I scaled to fit my NTSC comp
Remember what I said in the first post. You can't scale video with single pixel line detail (Most of the text in your first example) down to half it's size or less without seriously degrading the image. There just isn't enough data. AE, Photoshop, or Final Cut is simply not the right tool for the job. You need to use an app that scales the screen using the graphics processing unit (GPU or Display Adaptor) not an app that scales the pixels in an image.
If you need a DV sized image then first, work in Square Pixels, second scale the image in Screen flow. Set up your Canvas in Screen Flow, make you video and audio cuts there, then render your movie and bring that into AE or your NLE for final edit. You'll still have problems with the single pixel horizontal detail when you look at the image on a TV because interlacing in NTSC or PAL devices will not handle that fine detail without problems.
This example is scaled in Screen Flow. Notice how much better it looks? This text is too small to easily read, but it is not the mess that you'd get when using the wrong tool for the job.
The only better solution is to use a dedicated hardware screen recording device. With most training today delivered on the web and not on DVD the requirement to publish in NTSC format has all but disappeared. If your client or your project must be delivered on DVD in Standard Definition I'd try to move them to a web based delivery method or deliver the project in HD on Blue Ray. You'll have a much better project.