We have seen some Gaza Cs moving inside colonies, next to the ground where there should be normal gravity. Also, in the opening movie of one of the Gehren's Greed games a lot of Gaza Cs are seen landing on Earth. These accounts seem to support the idea that they can't operate on Earth.
And yet, at the end of the second Zeta movie, when Haman is checking the AEUG MS, she briefly grabs the Zeta by the shoulder. At that time it looks like the arms and torso of Haman's Gaza C are loosely connected to each other as each is constantly "moving" away of each other on their own, like they are not actually fixed together and which would make more questionable if such loose structure can actually function on Earth. Another common flaw I usually hear about the Gaza C si that its crude transformation system can only be used a limited number fo times, but at the time I can't find a reliable source for that claim so take it with a grain of salt. However, we are told that some of the surviving Gaza Cs used by the Sleeves are used like turrets rather than as MS (in episode 3 we can briefly see about 3 such units stationed on Palau's surface), so there might actually be some truth to such claim.
As for the Gaza D and Ga-Zowmn, they both seem to have learned from the Gaza Cs flaws and have better structures. In particular, by the end of ZZ there's one battle around Core 3 where several dozens of Gaza Ds are particiapting and there no longer seem t be Gaza Cs around, until Glemy's revolt probably leads to the remainign Gaza Cs back into active duty on both sides.
So, I would find more likely for the successors of the Gaza C to be able to operate on Earth. Furthermore, we are told that the Gaza E is in fact capable of flying in the atmosphere, and even better, capable of doubling as a sub-flight system, carry another MS on its back.
In the end, with the exception of the Gaza E, the Gaza types are mostly inteded to function in space, where they can make better use of their capabilities and since Neo Zeon ends up producing several MS specifically for the Earth invasion, it would seems unnecessary to deploy all those space use Gazas on Earth anyway.
As far as numbers go, both the Zssa and Dreissen were produced in quite large numbers, and while not as common as the Gaza types, we do see more than once large squads of both type of MS.