the Moon is too far away to see even large bits of it with the best telescopes currently available.
http://www.howstuffworks.com/question188.htm
various nations (not just USA) have left a lot of stuff behind on the Moon (or near it)
List of man-made objects on the Moon – Wikipedia, the free encyclopedia
and US spacecraft have identified some of it.
TBH even at the height of the Cold War faking a Moon landing would have leaked out somehow, (even if it were more recently along with everything else that has leaked from Washington) it would have cost nearly as much to fake it as to actually go there and caused far more reputational damage to the USA than even if the Soviets had got their first.
Also things like rockets can be detected from many miles away so both USA and USSR knew exactly what spacecraft had been launched.
Ironically it was space exploration which helped divert the best brains and technology of the Soviets and USA (and to some extent still does) towards peaceful uses of technology, particularly miniaturised electronics and long distance telecommunication. in many ways it was the other way round, stuff like moon landings, rockets, spacecraft helped inspire creative works from Hollywood to Hilversum – from movies and TV to product design – especially if you grew up in the 70s/80s this was apparent. Also every boffin involved with spacecraft was one less person who could be building an atom bomb (although the skills involved are transferrable (same as with broadcast engineering and telecoms) and can be used for either, in those days engineers multi tasking on projects was less common.
