There are things called 'proxies'. The same way you can assess the crime rate of a certain area by the amount of littering by the side of the road.
You mean incorrectly?
Also, if I had to guess, proxies that show global warming to be bad are widely publicised, whilst proxies that show global warming not to occur are ignored.
You are mixing categories. One is about the observable universe, ie. "does global warming happen"? The other one is a value judgement, ie. "is global warming bad?"
Sometimes it feels hotter or colder than the mercury would indicate, after all.
QED, it is impossible to determine if the climate is changing.