Given series is 372, 823, 644, 582, 46?, 8?7

Here the series follow the rule that :
372 = 3+7+2 = 12
823 = 8+2+3 = 13
644 = 6+4+4 = 14
582 = 5+8+2 = 15
46? = 4+6+? = 16 => ? = 6
8?7 = 8+?+7 = 17 => ? = 2

So the missing numbers are 6 and 2

The pattern follows the palindrome sequence.

Here the digits 0 1 4 7 8 5 8 3 6 9    9 6 3 8 5 8 7 4 1 0   0 1 4 7 ...

So the next digit in the sequence is 1.

The given number series is 200, 200, 100, 300, 75, ?

It follows a pattern that,

200

200 x 1 = 200

200 / 2 = 100

100 x 3 = 300

300 / 4 = 75

75 x 5 = 375

Hence, the next number in the given number sequence is 375.

The given number series is 2, 16, 3, 81, 4, ?

2

2^4 = 16

3

3^4 = 81

4

4^4 = 256

Hence, the next number is 256.

The pattern is + 1*1,+ 2*2, +3*3, + 4*4,+ 5*5,+ 6*6, . . . . . . .
So missing term is 56+36=92

The sequence is 13*13,12*12,11*11,10*10,9*9,8*8, . . . . . .
So,85 is wrong term and must be replaced by 9*81

The letters move 1, 2, 3, ... steps forward.
The numbers increase by + 3, + 5, + 7, + 9....

The first letter moves two steps forward. The last letter moves one steps backward. The number series runs as follows :
1 x 1 + 1 = 2
2 x 2 + 2 = 6
6 x 3 + 3 = 21
21 x 4 + 4 = 88.

The first digit is formed by adding the numbers which are the results of powers of 2 increased by one and second letter is the corresponding alphabet to that number...last number is the next number to the first one...therefore

15=7+8 (i.e. 2^3)

O is corresponding alphabet to 15

and 16 comes after 15 

The pattern is 2x32=64, 3x24=72, 4x19=76

The next number = 5x15.6=78