Solution:

As shown in the picture below.
Cut 1 : Cut the cake across the diameter.
Cut 2 : Cut horizontally and you will get 4 equal pieces of cake
Cut 3 : Cut horizontally from other side and you will get 8 equal pieces of cake

Solution:

A cigar stuck in the wall

Solution:

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

Solution:

One Night Stand.

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

Solution:

f the last elf is taking some time to answer the question then it shall mean that the elf's before him are all wearing distinct coloured hats. However sufficient time shall be given to the last elf to give the answer.
If he views the similar coloured hats in front of him , he shall quickly tell the answer

Solution:

2 degree below zero

Solution:

Remove the same 2 matchsticks from each figure and move figure towards left 45 degrees, It forms the letters V W X Y.

Solution:

Fifteen minutes can easily be measure using these two hour glasses.

Step 1: He will start both the hourglass.

Step 2: The moment the eleven minute hourglass is empty, he will invert it.

Step 3: When the thirteen minutes hourglass is empty, he will invert the eleven minute hourglass.
In step 3, we will have counted thirteen minutes. Since we inverted the eleven minute hourglass in step 2, it started from fresh and was inverted just for two minutes (13-11=2). In this manner when it is reversed when the thirteen minute hourglass is finished, it will have two minutes of sand left. This time when the sand finishes, he will have measured fifteen minutes. (13+2=15)

Solution:

888 + 88 + 8 + 8 + 8 = 1000

Solution:

To do this, he must be pushing the cork inside the bottle and then taking out the coin.

Solution:

I am Spice

Explanation:
Spice is hot if you remove first two alphabets then it will become "ice", which is really cold.