If I make a sandwich of wood, thin plastic, carbon-fiber tow (cut to about a foot long) impregnated with epoxy to the right ratio, thin plastic, and wood, and clamp it together and leave it, will I end up with a structure similar to a pultruded carbon-fiber rod? Will it have anywhere near the compressional strength of a product like these ? ..I'm not the first to consider this.. 
No - Pretension can throw off the compression modulus
While the properties can be calculated using formulas (depending upon the levels of reliability and degree of design, as shown in other answers on this site), the net result can still be off due to pre-tension in the fibers.
- Composites handle tension well, but are terrible in compression. This is primarily due to fiber buckling under compression:
- When pultruded components are cured, they are cured while under tension.
- The pre-tensioning leaves a residual tensile load in the fibers. This tensile load can (not always) allow the fibers to absorb more of the compression load before they begin buckling.
The only way to be sure if this has any impact on your parts is to fabricate them using tension and without pre-tension, and verify the compression modulus and strength of each part. Only after careful testing could you verify if the parts were affected by pre-tension or not.
Many pultruded FRP sucker rods are used in oil well pumping. The only pretension is that load used to pull them through the manufacturing process. String design keeps them in tension ( steel rods in the bottom of the string, FRP at the top). I believe this is because the compression properties are relatively poor. For compressive strength , I doubt that you want to emulate a pultruded rod.
No, and it would be hard to guess how far off it would be
The main difference is that pultruded rods have a controlled fibre/matrix ratio while an ambient-cured Epoxy will be off, even if you mix properly, as you cannot assure a homogeneous wetting of the tow with hand lay-up techniques and no debulking/vacuum pressing step.
That said, it can work for your application, just don't expect it to be comparable to pultruded rods.