Why and how is rust forming on moon?
UPDATE:
Today I got a chance to read the abstract the original paper which made news everywhere that moon has rust on the side which faces the Earth.
The authors propose that upper atmosphere oxygen from the Earth reaches the moon. In their words
&quot;Oxygen delivered from Earth’s upper atmosphere could be the major oxidant that forms lunar hematite. Hematite at craters of different ages may have preserved the oxygen isotopes of Earth’s atmosphere in the past billions of years. Future oxygen isotope measurements can test our hypothesis and may help reveal the evolution of Earth’s atmosphere.&quot;
Widespread hematite at high latitudes of the Moon in Science Advances (open access

Let us split your query into three parts:
a) Will iron oxidize without water and form oxides?
Iron would happily burn in oxygen to form iron oxides. But we cannot call it iron rust. So iron can oxidize without moisture. In fact, iron powder may spontaneously burn in air.
b) Will iron form rust without water?
Rust is more of a semantic issue.  Iron rust may not have constant composition. As per the dictionary definition (OED) &quot;

A red, orange, or yellowish-brown substance which forms progressively
as a flaking, permeable coating on the surface of iron and its alloys
as a result of oxidation, esp. through exposure to air and moisture.

Therefore, rust is an product of ambient environmental factors that affect iron at (approx.) typical room temperature. Rusting in sea water might different than rusting on a mountain. Acidic gases such as carbon dioxide, sulfur dioxide, nitrogen oxides certainly accelerate this process. Iron rust can be green and it can be common brown version. Since, chemically rust is a hydrated oxide of iron plus its hydroxide, water's presence is a must. More importantly water is a necessary ingredient which works as mediator of the electrochemical cell that forms between oxygen and iron surface. If you search Google Scholar you would find tons of articles on the electrochemical mechanism of iron rusting.
Coming to the more interesting question of rust on the moon:

nasaspaceflight.com: Rust on the Moon. How is that possible without
oxygen and liquid water? &quot;But how can rust form far from water ice
deposits on a barren oasis devoid of oxygen?&quot; Axios: Researchers find
rust on the Moon &quot;Scientists were surprised by the findings because
rust requires oxygen and water to form on Earth.&quot;

Since you would know more astronomy than I do, wasn't moon a part of the Earth? I recall this from a documentary. I do remember seeing layers of &quot;rust&quot; in moutainous paths, so the oceans had a lot of iron which settled as a layer in mountains. If moon were a part of Earth, why is that surprising?
a) The question is what was the age of this rust?
Secondly, Mars is also &quot;rusty&quot;, its surface is highly oxidizing with a huge amount of perchlorate in the soil (there is a long story how that was discovered by the folly of analytical chemists). Who knows what is the moon soil like. What about the shower of radiation on the moon. That may promote oxidation as well.