Lighting – Is it possible to increase CRI of LED lights/strips by using different ones together

You're not missing anything. LEDs that have a high CRI is a relatively new market segment, so the selection is limited, and prices are high. LEDs use the same tricks as fluorescents to reach reasonable CRI levels-- multiple phosphors with different spectra excited by the lamp's primary EM emission. So the potential exists to equal or exceed the best fluorescents. Right now, the consumer LED market is focused on replacing lamps in non critical applications, and thus tend to be rather low color temperature and only modest CRI levels. This situation should improve in the future.

"High" CRI fluorescent tubes are available in color temperatures from 3000 to 6500K, with CRIs between 80 and 90, and at reasonable prices. Right now, this is probably still a good solution for many applications. If the performance of these lamps are not adequate, halogen lamps are the only reasonable alternative. LEDs are not quite there yet.

I'm not familiar what that product specifically, but the "amplifier" in this type of lighting setup generally only reads the input from the previous upstream light strip and provides power from its power source. Therefore, the amplifier draws no current from the upstream lights.

Instead of 2 power supplies, you could use one power supply larger enough to satisfy all of your power needs.

To answer your questions specifically:

1. Yes.
2. No. Power supplies are rated for the current that they CAN deliver. Using a power supply larger than needed is fine.
3. Again, not knowing these devices specifically, I am going to guess no based on my experience with other, similar products.
4. Generally no, provided you do not draw more power than the supply is rated for.