- yes, grounds must always be connected. But also take ground lines directly to the central PSU. Otherwise you will still get voltage drop and incorrect colours. Remember the current has to flow in a circuit, and the circuit is only as good as it's weakest link, which would be the ground lines, looking at your schematic.
- connect the grounds but use each PSU to power separate sections of LEDs. Don't connect the 5V lines from different PSUs.
- thin wires will cause voltage drop and incorrect colours. Use thicker wires. Connect 5V to both ends of each strip to reduce voltage drop further.