Electrical – Why doesn’t this LED strip work, even though the power supply seams appropriate

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.

No, you can't mix and match 3528 strips/connectors with 5050 strips/connectors.

Even within 5050, RGB vs RGBW use different connectors, and there are even W-only strips in the 5050 form factor, which use a different connector still.

Put more of an effort into shopping, and you should be able to find the right items more easily. EBay/AliExpress are not necessarily the best places to shop.