The problem is that array.index() will return the first instance of a value. You have the value 84 twice - so since the first index is odd, you never add it.
You really need to keep track of the index, not rely on uniqueness of the values. You do this with
for idx, val in enumerate(array):
now your first value will be the index, and the second value will be the value. Test idx%2==0 and you can figure it out from here.
update here is the complete code, making clear (I hope) how this works:
checkio = [-37,-36,-19,-99,29,20,3,-7,-64,84,36,62,26,-76,55,-24,84,49,-65,41]
def checkzi(array):
    if len(array) != 0:
        sum_array = 0
        for idx, x in enumerate(array):
            print "testing element", idx, " which has value ", x
            if (idx % 2 == 0):
                sum_array += x
                print "sum is now ", sum_array
            else:
                print "odd element - not summing"
        print (sum_array)
        answer = (sum_array) * (array[len(array)-1])
        return (answer)
    else:
        return 0
checkzi(checkio)
Output:
testing element 0  which has value  -37
sum is now  -37
testing element 1  which has value  -36
odd element - not summing
testing element 2  which has value  -19
sum is now  -56
testing element 3  which has value  -99
odd element - not summing
testing element 4  which has value  29
sum is now  -27
testing element 5  which has value  20
odd element - not summing
testing element 6  which has value  3
sum is now  -24
testing element 7  which has value  -7
odd element - not summing
testing element 8  which has value  -64
sum is now  -88
testing element 9  which has value  84
odd element - not summing
testing element 10  which has value  36
sum is now  -52
testing element 11  which has value  62
odd element - not summing
testing element 12  which has value  26
sum is now  -26
testing element 13  which has value  -76
odd element - not summing
testing element 14  which has value  55
sum is now  29
testing element 15  which has value  -24
odd element - not summing
testing element 16  which has value  84
sum is now  113
testing element 17  which has value  49
odd element - not summing
testing element 18  which has value  -65
sum is now  48
testing element 19  which has value  41
odd element - not summing
48
You obviously want to take the print statements out - I added them to help explain the program flow.