"Did Star Trek and Lost In Space lift some ideas from Forbidden Planet?"
Apparently, there are many elements in "Forbidden Planet" that 'inspired' Gene Roddenberry, but where he borrowed too much, IMHO, is revealed by the mind power stories we've seen in both pilots for Star Trek, i.e. "The Cage" and "Where No Man Has Gone Before".
In the field of treknological speculation there is of course one difference: Franz Joseph Schnaubelt liked "Lost in Space" but was no fan of Star Trek.
IMHO, he should have been doing the blueprints for the Jupiter 2
from "Lost in Space" and then, some passionate Star Trek fan should have taken the inspiration to do blueprints of the Enterprise
and a technical manual.