So. On the subject of kaiju and Rangers and Zords (oh my).
The only real issue I'd have with the Power Rangers running around would be that it makes G-Force and their own kaiju-hunting teams more or less irrelevant, which would scrap an awful lot of things I had in mind. However, if the Rangers and G-Force sort of operate the same way that the Avengers do in regards to SHIELD, the one wouldn't necessarily overlap the other.
As for the matter of other giant-size heroes and monsters making Godzilla less special, well, scale is a very important thing to keep in mind.
Looking at some of the Super Sentai wiki pages, it looks like the Megazord, the Tiger-Zord, and most of the other bipedal Zords stand around 50 meters tall and weighing in at around 150 metric tons, which puts them in the same range as most of the kaiju in the Showa era of movies ('54-'75). To use Pacific Rim's kaiju-classifying system, most of them would be a Category 2. (Rodan's offensive capabilities and Anguirus's heavy armor might make them Cat-3, so before Goji arises, they'd be primary rivals for 'king' of the island). Given the firepower the Rangers are loaded with, they'd probably be able to knock most of them around with little effort.
Godzilla himself, though, is a whole 'nother story. I'm using the Heisei ('84-'95) incarnations of Godzilla, as well as Mothra (and when he shows up, King Ghidorah), and they're all roughly twice the size of their Showa counterparts. Heisei Goji stands at over 100 meters tall, and weighs in at over 60,000 metric tons. That makes him an easy Category 4, and later Heisei monsters like their series' Ghidorah, or Space Godzilla or Destoroyah, would be well into Category 5.
So even the most pimped-out, final-form Megazord would barely come up to Godzilla's waist, and only carries about 1/500th the mass of the King. Even with the Rangers' high-tech and magic weapons, they really don't have even remotely the kind of power needed to slug it out with Big Green in a straight-up fight.
Though that in and of itself would make for a much more interesting story than just "two big things punch each other until one of them leaves," since the Rangers would have to fight a lot smarter against Godzilla than they would against basically anything they've ever encountered before.
