Solution:

'T'

Explanation:
The letters around the center are actually depending on the letter in the center. They are advancing according to the numerical value of that letter.

For example, F, G, H and I are advancing by 1 because the center letter is A, which has the numerical value 1.

Again, G, K, S, and O are advancing by 4, because the center letter is D, which has a numerical value of 4.

Similarly, N has advanced by 6 from H, because the center letter is F, which has a numerical value of 6. Hence, the missing letter should be T.

Solution:

She is near the trunk of the tree as shown below.

Solution:

Anybody will prefer the shop in the eastern market for sure. There are two reasons for it:
Firstly, since there are only two barbers in the town, the one running the shop in eastern market must have gotten his hair done by the one running the shop in the northern market and vice versa. Thus, if the barber in the eastern market has a shabby haircut, it must be due to the incompatibility.
Secondly, the neat and clean shop can be due to the scarcity of customers as well. While the untidy place refers to the fact that so many customers get their hair done in that shop.

Thus any one will opt for the barber shop in the eastern market.

Solution:

40 balls

Solution:

The Password.

Explanation:
Clues 2 and 5 are correct and rest are lies.

Solution:

Treasure Map Is Hidden Inside The Hanging Room Floor.

Explanation.
Just reverse the text "roolF mooR gnignaH ehT edisnI neddiH sI paM erusaerT".

Solution:

2100 miles.

Each vowel equivalent to 500 and each consonant equivalent to -200

Solution:

zero

I had 3 green socks and 1 Pink sock.

=> chances of pair of green socks = 1/2
chances of pair of mixed socks = 1/2
chances of pair of green socks = NULL

Solution:

A Question

Solution:

Old password : altered

New Password : distinct

Solution:

The Lady sitting in the chair has the detector attached to the back of her blouse.