Solution:

10992

Explanation:
Each girl has seven bags with seven adult cats each.
Also, each adult cat has seven little cats.
Thus each girl has 49 adult cats and 343 little cats.

Hence, seven girls will have
343 adult cats and 2410 little cats.

If we calculate the number of legs now.
Girls = 7 * 2 = 14
Cats + Little cats = 2744 * 4 = 10976

Total legs = 10976 + 14 = 10990

Since I also walked in the room, there will be two more legs.
Now total legs = 10990 + 2 = 10992.

Solution:

Birbal drew another line that was larger than the pre-drawn line. Now the pre drawn line was smaller.

Solution:

One Minute building

building 62 => Sixty second building => One minute building

Solution:

The code word is TIE.

Let us assume that the word is MOD. Now if you are told one character of this word, you would not be able to identify the number of vowels in the word. For example if you are told M, then the word can be MOD or AIM but they both have different number of vowels. If you are told about O, then there are two words MOD and OAT which again have different number of vowels. If you are told about D, there are two words again MOD and DUE which have different numbers of vowels again.

In such manner, all the words that have M, O or D in them can be excluded. Then only word that remains is TIE.
If you are told about T, there are two words OAT and TIE which both have two vowels.
If you are told about I, there are two words AIM and TIE which both have two vowels.
If you are told about E, there are two words DUE and TIE which both have two vowels.

Thus TIE is the code word.

Solution:

At every floor, the detective had to open the window himself. If the man had committed suicide, then at least one of the windows must have been open. A person who is committing suicide won’t close the window behind him. Thus he knows that someone pushed him off the edge and closed the window afterwards.

Solution:

The password is U472BMT.

Read aloud the line and you will sound something like "U Four Seven 2 B M T".

Solution:

It can be down as shown below.

Solution:

Adam and Eve were the only people without navel. As they were not born out of a mother, they never had umbilical cords and thus never had navels.

Solution:

Yes, some people are going get really tiny pieces but you were allowed for that right. Also, there is another way of doing it. Can you think of it?

Solution:

The women is a of short stature. She can't reach the upper elevator buttons, but can push is with his umbrella.

Solution:

The competition was held on 11 November 2013. The logic is simple. In America, the dates are mentioned as Month/Day/Year and in Britain, dates are written as Day/Month/Year. But on 11 November 2013, the date and month will be the same. And thus both will write in the same manner i.e. 11/11/2013. If it would have been any other date, the evaluators would have been able to identify it.