That's not breadboard compatible either. It's a surface mount chip. Look for a DIP chip. These have pins with 0.1" spacing, the same as a breadboard.
No, you need both. A ground wire running alongside the data wires between the strips. This does not need to be heavy gauge, it will not carry much current. But it will prevent induced voltages corrupting the data signal. You also need the ground lines back to the PSU also. These should be heavy gauge, they will carry the high currents.
No, the esp board will have caps built into it already. The bypass cap is for the chip you will use as a level shifter.