Roth IRA retirement accounts are supposed to be funded with earned
income.  Is there any precedence or official explanation on how young
a child can be and begin making contributions to a Roth IRA account?

For example, if a person pays their four year old child $333 per month
to take out the trash, can that $4000 per year be put into a Roth IRA
for the child?

ON another question, someone wanted to pay his wife for tax reasons. 
The answer was that the IRS doesn't go along with that, so I expect
"paying" one's child would be considered a gift, especially since I
assume that the kid would not be allowed access to the funds.

As to the first question:  a child actor or model would indeed earn
money, regardless of age.

You can pay your child if you have any kind of business at home.  If
it would be reasonable to pay someone else the wage to perform a task
then you can hire your child instead and pay the reasonable wage.
http://www.lifetimefp.net/Advice090301.htm

Sorry,  jack_of_few_trades-ga.  Go directly to jail.  Do npot pass go.
 Do not collect $200.  That would be child  labor.  (The kid is 4.)

"Minimum Age for Employment:
The minimum age for employment is 14 years old. There are some
exceptions such as newspaper delivery; performing in radio,
television, movie, or theatrical productions; and work for parents in
their solely-owned nonfarm business (except in manufacturing or in
hazardous jobs)."
http://www.stopchildlabor.org/USchildlabor/fact1.htm

But do check your state child labor laws as they may be more strict
than the US laws.

If you solely own the business (such as most people who run their own
business from home) then your child is fair game as far as US laws go.
 Hire him to wash the car if you drive at all for work, clean your
office... be creative, what else can he do or help with that you could
pay someone else to do?

There's no free lunch in Roth IRAs. Roth IRAs were designed to
accelerate tax payments to the government and they do. (The money is
taxed on the way in.) Whether or not a Roth is good for you (or your
child) depends on whether you (or they) are in a higher (good) or
lower (not good) tax bracket when you take the money out. The real
value of a Roth is very difficult to predict when you're a long way
from taking the money out.

Govt, since the question is about a child (with no earned income at
the moment), the tax rate is currently as low as it ever will be.  The
chance that this person will have a lower tax rate in retirement than
now is about as close to 0 as possible.

Also, taxes today in the US are about as low as they have ever been. 
I (and almost any expert out there) think that taxes will be on the
rise over the next couple decades to pay for all of our old folks and
the tremendous national debt.  With this in mind, most people are in a
lower tax % today than they will be in retirement.

If interest is allowed to compound the income would eventually rise to
the level of taxability.  In a Roth it would compound fully.  Granted
at this level the difference is not great, but there would be one.