Where does the belief that the Earth is relatively young (6000 years) come from? Some christians believe the Earth is only 6000 to 12,000 years old. Where does this age originate from?
No, this is not a universal belief. Some do not take the days in Genesis as literal, and some treat Paradise (as described in Genesis) as being a spiritual world rather than being the same 'world' which we measure with carbon dating and other physical measures.

The 6000 years is roughly discerned using clues from stories in the Bible, but there are other measures placing its age closer to 7500 years.

What is generally used is the following:


Length of 'day' in Genesis 1
Length of time figures in Genesis lived
Ages are added by using 'and in so-and-so's xth year, so-and-so was born' (and variations thereof)
History / chronicle books have king-dynasty lengths.


It's a lot of work to do, really.

Variations in measures can be explained by differing interpretations of 'day' in Genesis, whether 'in the xth year of y' means that x years preceeded it or x-1 years, and estimates of times not explicitly mentioned.