Solution:

The timer of a microwave.

The timer of a microwave generally consists of 4 digits. First two digit for minutes and next two for seconds.
So,
99 indicates - 1 minute 39 seconds
100 indicates - 1 minutes (1:00)

Solution:

One Minute building

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

Solution:

28 =>1+2+4+7+14

cannot be 1 (as number cannot be included)

Solution:

Steps:
1) Light fire on first bar from both the ends and second bar from one end.

2) After 30 seconds(when first bar gets completely burned out) , burn the second bar from second end as well.

3) When second run completely gets burned , you know its 45 seconds.

Solution:

Everyone will done one day or other.

Abraham Lincoln era is too old and we expect most of them(if not all) to be dead by now.

Solution:

White should move the rook from
A1 to D1 and hence complete the castling move.

Solution:

2003.

Explanation:
Its a series of prime numbes.

Solution:

It will take the man 58 minutes to reach the top.

Since he is climbing 3 feet and falling 2 feet back every passing minute, he is actually climbing 1 feet every minute.
Therefore, he would have climbed 57 feet in 57 minutes. The remaining distance is 3 feet. He will take one more minute to climb it and reach the surface. Since he has reached the top surface, he won't slip back.

Solution:

Sansa has the siamese (conjoined) twin sister.

Solution:

It can be solved as below.

Solution:

19

Explanation:
The children can purchase 19 chocolates in return.

Out of 77 wrappers, 75 will be used to buy 15 chocolates and two will be left spare.
The 15 chocolates will create 15 empty wrappers that can be exchanged to get three chocolates.
Three chocolates will return three wrappers which will help them buy another chocolate.
Now the wrapper from this chocolate and the two spare that were left earlier will get them another chocolate.
15 + 3 + 1 = 19