I think the answer to the first question is:
11
Reasoning
All sequences deal with differences that are divided by 2.
4 + 4 = 8 + 2 = 10
9 - 4 = 5 - 2 = 3
20 - 6 = 14 - 3 = 11
Update
I think the answer to the second question is:
13
Reasoning
Part 1
Exclusion. BOY can't be 10, because then it would be GIRL - BOY = BOY. And that's not correct obviously.
It can't be 14 either, because then POOR would have to be 6. If you take the lowest possible numbers (higher than 0), you would get 211?.
Where ? is the value for R, which can't be 2 or 1, meaning you get a result higher than the 6 you'd need.
Part 2
Exclusion. The higher values 67 and 98 would mean POOR would be negative.
That would mean that if you take the P as the minus (-), you still would need the 3 other letters to contain enough to add up to 78, which is the highest possible difference using 98. Adding up the highest possible values in that case , 998 = 26, doesn't come close to 78. Also, assuming a possitive number, a fictional word of 4 equal letters can only contain 9999, 9+9+9+9 = 36.
Part 3
Exclusion. 2 isn't possible because: Including the use of 0 (zero), you would get BOY = 0 + 1 + ?, where ? can't be 1, so you would not get 2.
Part 4
20 - 13 = 7
POOR = 7 -> 3112
BOY = 13 = 517
GIRL = ? + ? + 2 + ? = 4 + 8 + 2 + 6
This is 'a' solution, which further confirms it should be 13. The actual way of decoding the letters doesn't matter.