There are two ways to measure the GDP of an economy, right?  A flow of
product and earnings approach, right?  Well, what I still don't
understand is how double counting is avoided in each approach and why
the level of GDP (should be) the same in each approach.
Furthermore, is there anything that GDP doesn't measure?

There are three ways to measures GDP:

1. Income method
2.Exprnditure method
3. VAlue Added

VAlue Added- Double Counting is avoided by 2 ways:
a. Either taking ONLY the value added by each sector. ex. 
seeds>wheat>bread

Now we take the VALUE added in the Wheat & the value added in bread
only. Therefore the problem of double counting is solved.

b. IF we take the value of only the final goods i.e. bread in the
above example then also this problem is solved.


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Using both Income and expenditure approach, amount of GDp will be same
because according the argument of Circular Flow of Income: Every
payment (expenditure) made by a buyer becomes income for the seller.
Therefore, the dollor value of the purchases (exp.) of new goods and
services in a year must be equal to the dollor value of the income
generated in that year.

GDP measure the following things:

1. Value of leisure
2. Illegal activities
3. Social welfare (transfer payments)
4. Goods produced in the past years

IN the above comment please note that GDp DOES NOT include those things