No. The dust on the moon is accumulated micrometeorites.
~ 50 to 60 tons / day
About 100 tonnes of "micrometeorites" litter Earth every day.
That depends how big an "object" you want to accept. Lots of micrometeorites - pieces of dust, basically - fall on Earth each day. The Wikipedia article on micrometeorites estimates about 1017 particles larger than 10 µm a year. That's many millions every second.
Micrometeorites and undetectable bits of space junk (0.4 mm small) pose serious threats to every current and future manned space mission. The particles could travel in as fast 12 miles per second, with enough momentum to melt and vaporize aluminum spacecraft skin.
The major cause of mechanical weathering on the moon is micrometeorites. These tiny rocks hit the moons surface and over time, change it.
The footprints on the moon left by the astronauts will still be there in a few thousand years. They will eventually be covered with new dust from micrometeorites.
Because they are traveling at high rates of speed and there is no atmosphere to slow them down before they could contact people or craft, which could cause death or damage.
The Earth is steadily getting heavier because thousands of tons of meteorites and micrometeorites fall onto its surface every day.
No atmosphere or water to do any weathering. The weathering of the moon largely consists of being hit by micrometeorites, mostly dust-sized.
The basic requirements would be any material that could be made airtight and heat insulated. In addition it would need to withstand solar flares and micrometeorites.
Meteors (meteorites, actually) come in all sizes, from microscopic to bigger than houses. One can make artificial categories of 1) micrometeorites (very common); 2) gravel-sized meteorites (less common, but affordable) and 3) massive (seen only in museums).