Inversion count via divide and conquer
1. Bug

If you pass the empty list to Sort_and_Count, it recurses until it runs out of stack:

&gt;&gt;&gt; Sort_and_Count([], 0)
Traceback (most recent call last):
  File "cr107928.py", line 67, in test_Sort_and_Count
    Sort_and_Count([],0)
  ... many lines omitted ...
  File "cr107928.py", line 49, in Sort_and_Count
    if n==1:
RecursionError: maximum recursion depth exceeded in comparison


2. Review


The code omits spaces around operators. This makes it hard to read. It is better to follow the Python style guide (PEP8):


  Always surround these binary operators with a single space on either side: assignment (=), augmented assignment (+=, -= etc.), comparisons (==, &lt;, &gt;, !=, &lt;&gt;, &lt;=, &gt;=, in, not in, is, is not), Booleans (and, or, not). [...] If operators with different priorities are used, consider adding whitespace around the operators with the lowest priority(ies). 

The names do not follow the Python style guide. This recommends:


  Class names should normally use the CapWords convention.


so TestSortAndCount would be better than TestSort_and_Count.


  Function names should be lowercase, with words separated by underscores as necessary to improve readability.


so merge_and_count_inversions would be better than Merge_and_CountSplitInv.

It's not compulsory to follow the Python style guide, but it makes it easier for other Python programmers to read your code.
Python's documentation uses the term "sequence" to refer to indexable collections, not "array". There's less risk of confusion if you stick to the standard term.
The docstring says, "Merges two similar length pre-sorted sub-slices", but in fact the function works fine even if the subsequences are not of similar length. I know that when called from Sort_and_Count the subsequences will be equal in length or differ by one, but it's better to document the actual behaviour of the function.
Instead of index_1st and index_2nd, I would use i and j. These variables are used many times, so using short names reduces the amount of code and make it easier to read.
The arguments len_1st and len_2nd are only used within the expressions start_index+len_1st and start_index+len_1st+len_2nd. This suggests that the arguments to the function should be start, middle and end instead.
Instead of updating A one element at a time:

write_index=start_index
for value in temp_array:
    A[write_index]=value
    write_index +=1


update all elements at once using a slice assignment:

A[start_index:start_index + len(temp_array)] = temp_array


This is efficient (in CPython, anyway) because the replacement is the same length as the slice being assigned.
In Sort_and_Count, the caller is expected to pass in the length of the sequence. It would be simpler if Sort_and_Count called len instead. This could easily be done using a helper function; see updated code below.
The test case doesn't try the empty list (see "Bug" above).
The test cases only test that the inversion count is correct, but don't actually test that the sequence is sorted! You need something like:

self.assertEqual(A, sorted(A))

The test cases have hard-coded counts of inversions. But it is easy to count the inversions in a sequence, using itertools.combinations:

sum(j &lt; i for i, j in combinations(seq, 2))


This is inefficient (it's \$ Θ(n^2) \$), but clearly correct, and so a good comparison for the efficient, but not clearly correct, counts produced by the function under test.
The test cases are very repetitive. This could be avoided with a helper method, something like this:

def check(self, seq):
    expected = sum(j &lt; i for i, j in combinations(seq, 2))
    found = sort_and_count_inversions(seq)
    self.assertEqual(expected, found)
    self.assertEqual(seq, sorted(seq))

The test cases are not very thorough. You could easily test many more cases using random test generation:

for _ in range(100):
    self.check([randrange(100) for _ in range(randrange(100))])



3. Revised code

from itertools import combinations
from random import randrange
from unittest import TestCase

def merge_and_count_inversions(seq, start, middle, end):
    """Merge two adjacent sorted sub-sequences of seq and count
    inversions.

    Arguments: 
    seq -- sequence to sort
    start -- index of beginning of first sorted subsequence
    middle -- end of first, and beginning of second, sorted subsequence
    end -- end of second sorted subsequence

    Result: number of inversions (cases where i &lt; j, but seq[i] &gt; seq[j]).

    Side effect: seq is sorted between start and end.

    """
    assert 0 &lt;= start &lt; middle &lt; end &lt;= len(seq)
    inversions = 0
    temp = []
    i = start
    j = middle
    while i &lt; middle and j &lt; end:
        if seq[i] &lt;= seq[j]:
            temp.append(seq[i])
            i += 1
        else:
            temp.append(seq[j])
            j += 1
            inversions += middle - i
    if j == end:
        # Second subsequence is complete: process remainder of first.
        temp.extend(seq[i:middle])
    else:
        # First subsequence is complete: no need to process
        # remaininder of second, since it does not move.
        pass
    # Insert sorted results into original sequence.
    seq[start:start+len(temp)] = temp
    return inversions

def sort_and_count_inversions(seq):
    """Sort a sequence and count inversions.

    Argument: seq -- a sequence

    Result: number of inversions (cases where i &lt; j, but seq[i] &gt; seq[j]).

    Side effect: seq is sorted.

    """
    def sort_and_count(seq, start, end):
        if end - start &lt; 2:
            return 0
        middle = (start + end) // 2
        return (sort_and_count(seq, start, middle)
                + sort_and_count(seq, middle, end)
                + merge_and_count_inversions(seq, start, middle, end))
    return sort_and_count(seq, 0, len(seq))

class TestSortAndCount(TestCase):
    def check(self, seq):
        expected = sum(j &lt; i for i, j in combinations(seq, 2))
        found = sort_and_count_inversions(seq)
        self.assertEqual(expected, found)
        self.assertEqual(seq, sorted(seq))

    def runTest(self):
        self.check([])                    # empty sequence
        self.check([1])                   # single element
        self.check([1, 3, 5, 2, 4, 6])    # even length
        self.check([1, 3, 5, 2, 4, 6, 3]) # odd length, duplicate value
        for _ in range(100):              # random test cases
            self.check([randrange(100) for _ in range(randrange(100))])