Forbidden Planet was a watershed moment in science fiction. It had a tremendous influence on jsut about everything to come after it, so it's not unusual that Rodenberry would insert little homages to it in Trek. The 1701 thing, and the United Federation of Planets are probably something along those lines.
Most of the other stuff you mention are things you will find in many many movies. It's more of a case of both having similer influences than one being based on the other.
Wireless communication. Faster than light travel. Saucer shaped space vehicles. The Captain getting the girl. None of these things are unique to the Forbidden Planet.
The Captain/First Officer/Doctor relationship exsists in almost every naval themed picture ever made. And if you change the ranks a bit you could extend that to most military movies and westerns too. The three-character dynamic is a classic form in literature. Heck, you could, and many have tried too, make the case that it's a platonic form of the love triangle. That's a little wierd for me though, so I'll stop right now.
