Passing on the DNA of old age
It is good thinking you're doing here!

Only mutations happening in the germ line can be passed on

Mutations that happen in an individual hand (or any other soma line) will not be passed on the the offspring. Only those happening in the soma line can.

Mutations are random

What I mean here is that it is not because someone is old that all of sudden the mutations that would happen in this person would be beneficial for old people.

Age dependent "quality" of genome being passed on

You are asking 


  Do older parents pass on a different, perhaps more learned "quality" of DNA?


You seem to misunderstand how natural selection works. An individual alone does not get selected over its lifetime to become better (at least not in the sense of what is meant by "natural selection"). You need a population for selection to happen.

In the females as the pool of ovules is being set early in life. In males the spermatozoids keep replicating during the whole lifetime of the individual. Hence, in males, but not in females, the number of new mutations being passed on to their offspring should increase with age and it does.

In humans the mother transmit on average 15 new mutations and the father transmit on average $25+2(g−20)$ new mutations, where $g$ is the age of the father (Cochrane and Harpending, 2013; this approximation holds only for men older than 20 years of course).

As the vast majority of mutations are neutral or deleterious, the "quality" of the genome being passed on should decrease with age unlike you expected.

Efficiency of selection decreases with age

The fact that after a given age, one does not reproduce anymore (among mammals, this is particularly true for women) does affect the selection pressures on the genetic variants that affect age specific fitness. e.g. A mutation that will reduce health at young age will be quite strongly selected against while a mutation that affect health at old will not.

When talking about this discrepancy between selection pressures at different ages, the concept of antagonist pleiotropy often comes into the discussion. Pleiotropy is the phenomenon where one locus affect several traits.

The idea of age-specific antagonist pleiotropy is that a mutation that will increase health at a young age by a factor $s$ but decrease it at an old age by the same factor $s$ will be selected for. Such process can cause a population to evolve toward lower lifespan. The antagonistic pleiotropy hypothesis is one of the most important hypothesis that we have for the evolution of senescence.

Evolution of senescence 

You might want to have a look at the posts:


Why does evolution not make our life longer?
Why do we age? or Do we have a theory of senescence?