Can a gene-expression or epigenetic 'user-history' be found in the body?
First Paragraph - let's debunk what was poorly phrased

Many sentences you wrote are quite misleading. You should have a look at an introduction to molecular genetic to have a better understanding such as what you can find on Khan Academy. In short:


  the double helix DNA strands are the building blocks of an organism


This sentence is very opinion based. as a matter of proof, one could for example replace the term double helix DNA by "cells" (or other terms) and many would still be satisfied with the sentence.


  Genes are chunks of DNA telling the body what to do.


A gene is a part of DNA which contains regions coding for proteins. There are many sequences which are not coding for any proteins that are essential for the individual survival (such as regulatory sequences or sequence coding for rRNA for examples).


  Genes naturally switch on and off.


The regulation of gene expression is actually much more continuous than just on/off. Also, regulation of protein concentration can occur at different levels and not only by regulating gene expression.


  DNA is essentially stored in Chromosomes.


This sounds like chromosomes are boxes in which one can find DNA. In reality, DNA is a type of molecule which can be found under different forms. In humans (and other eukaryotes) most of the DNA (but not all) is present in the form of chromosomes.

Second Paragraph - What is epigenetics?


  External and environmental factors can influence the expression of genes,


This is correct


  and the information stored in the DNA


This is wrong. Environmental factors do not affect the sequence of base pairs of the DNA. It only regulate its expression. Such regulation of expression can last for a long time and even be transmitted to the offspring. The term epigenetics is loosely defined. In short epigenetics refers to the set of mechanisms that affect gene expression at short and long term. The classic mechanism of interet in epigenetics in the methylation of histone tails (quickly speaking, histones are proteic structures around which DNA is wrapped). 

Your question


  Does the body store a history of these expressions? Do the chromosomes (or some other part) of an older organism store a 'user-history' of which genes where previously activated when the organism was younger? eg, can you tell which genes where active when a person was 12, from the cells of an 80 year old?


As explained above, some of the molecular mechanisms yielding to changes in gene expression can have long term consequences on gene expression (which can even be inherited). But there is not data bank (or other centralized or not source of information) that specifically record the expression of each gene at any one time in the history of the cell life.