Pandas Melt with Multi Index Data Set and Resetting Index - Why is this working?

I created this dataset straight from the pandas documentation:

In [28]: columns = pd.MultiIndex.from_tuples([('A', 'cat'), ('B', 'dog'),
   ....:                                      ('B', 'cat'), ('A', 'dog')],
   ....:                                     names=['exp', 'animal'])
   ....: 

In [29]: index = pd.MultiIndex.from_product([('one', 'two'),
                                             ('bar', 'baz', 'foo', 'qux')
   ....:                                     ],
   ....:                                    names=['first', 'second'])
   ....: 

In [30]: df = pd.DataFrame(np.random.randn(8, 4), index=index, columns=columns)

The MultiIndex dataset (for both columns and rows) looks like this:

I wanted to get to something like this [the image is truncated but you get the point]

There are probably a zillion ways of reshaping this but I want to get it done using unstack() and melt()

These are the two ways I came up with:

    1. pd.melt(df.reset_index(),id_vars=['first','second'])
    2. pd.melt(df.unstack().reset_index(),id_vars=['first'])

So here is where I am stuck: Why does this work?

df.reset_index() gives me this dataframe

with these columns

'first' and 'second' don't show up in the names of columns. They're are infact levels of the column exp. So I wondered what'd happen if I added more levels to the id_vars in the melt

If I change the melt to

pd.melt(df.reset_index(),id_vars=['first','second','A'])

I get the following error:

ValueError: arrays must all be same length

If I change the melt to

pd.melt(df.reset_index(),id_vars=['first','second','dog'])

I get the following error:

KeyError: 'dog'

Can someone explain what's intuitively going on under the hood with the reset_index() and why doesn't melt accept other levels? Why are 'first' and 'second' showing up as levels instead of columns?