Trace vs Plane on Multi-Layer Board
At connectors narrow traces are used to connect terminals to the planes to  allow wave or hand soldering to heat the joint sufficiently,  if oven reflow soldering is used instead these thermal releifs are not needed.
as the thermal releifs are symmetrical they do not have a dipole moment and thus do not radiate significantly.

There's nothing that two ground planes can do that one ground plane and one correctly bypassed power plane cannot do better.

if you need to split the power plane into regions for different voltages that can work well too.