Do I need to add the ground wire from the transformer ground lug to a common ground connection of my station?
The ground lug at the transformer bottom is meant for use with a dedicated ground rod when the transformer is installed up to 10 feet (3 meters) above ground. If it is not practical for you to add a dedicated ground rod for an EFHW antenna, just omit that connection and ground the coax shield of the feed line to a station ground, entry connector plate, or dedicated ground rod with pass-thru SO239 or with lightning arrestor at the entrance to the home/radio room. The grounding of all coaxial cable shields at the entrance to the home is required by National Electrical Code NEC 810
Also, do not listen to internet “experts” saying that some form of hanging counterpoise wire of 0.05 wavelength should be added to the ground connection of the transformer box. It is useless and all you need is to provide good grounding of the coaxial cable shield anywhere along the line between the transformer and entrance of the cable to your home.
The EFHW antenna will work without any grounding too but it is recommended for protection of your equipment.
As original article of an inventor Josef Fuchs, published in July 1928 QST magazine says this:
“The unilateral connection of the aerial to an intermediate circuit as seen in Fig.1 has the advantage of working without ground or counterpoise…”