Solution:

Congratulations

Explanation:
Message Was: Company Piano Numb Dining Racial Mozetta Traitor Meou Legacy Quintana Trick Missi Orange Frozen Sapphire

Pick the 1st letter of the first word, 2nd letter of the second word, Pick 1st letter from the third word and so on.

Company - C
Piano - O
Numb - N
Dining - G
Racial - R
Mozetta - A
Traitor - T
Meou - U
Legacy - L
Quintana - A
Trick - T
Missi - I
Orange - O
Frozen - N
Sapphire - S

Solution:

XIX is the answer which refers to nineteen in roman numeric system. On taking out one i.e. I, we will get XX that is equal to twenty.

Solution:

One Night Stand.

Explanation:
Number 'One' + Knight(chess piece) + st + and(symbol &)

Solution:

Both will rotate at Same speed.

Explanation:
Gears of the same size will always rotate at constant speeds despite any smaller or larger gears in between them.

Solution:

Pages 19, 20, 23, 37, 38, & 42 will also be missing

Solution:

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

Solution:

There is no mention if you can use sign or not and thus the only possible way to get this is as follows:

(15 - 9) + (13 - 7) + (7 - 1) + (9 - 1) + (13 - 9)

If you solve inside the brackets, you will get the following equation
6 + 6 + 6 + 8 + 4
Adding all these numbers, you will get 30 which is the required sum.

Solution:

What is the correct time ?

Solution:

For this answer is 3^0, 3^1, 3^2... That is 1,3,9,27,81,243 and 729.

Solution:

The teacher will use a simple trick to form a perfect impression. He will ask all his students to raise their hands on each of the question that is asked. But the only catch will be that those who knows the answer correctly will raise their right hand and others will raise their left hand.

In this way, the principal will see all the hands being raised for each question even though all won't be knowing the correct answer. The teacher will ask only those who know the answer and they will always be correct. So the principal will be impressed to full extent.

Solution:

16-days

In 15-days, he will climb 15-inch and when he climbs 5 inch more, he will be out of the well :-)