1) What kind of element(s) make up the egg shell?

Question

2) Mammals including human have eggs, but why they (their eggs) don't come in the same way as some other animals like birds, or chicken ?

Answer 1

Thats because humans gestate (hatch) their eggs internally. A chicken sits on an external egg. Most shells are made of calcium carbonate or some other calcium based compound.
Chickens are not mammals, and thus do not have a placenta to nourish the embryo. All a chicken's nourishment comes from a yolk. This is true for most reptiles and amphibians as well.
All mammals gestate their young internally, with a placenta. The embryo is sort of a temporary parasite taking nourishment from the female's blood stream via the placenta. After birth, mammals live on the milk of the mother until they are able to eat solid food.
The only mammals that lay eggs are monotremes, which include platypuses and echidna (a small vole-like critter). These mammals lay eggs with leathery shells reminiscent of skin. They nurse their young after birth like all mammals do, with milk.

Answer 2

The primary material in a bird's egg is calcium.

The form an animal's egg cell takes is related to the environment in which that cell will grow. A human cell requires no shell because it grows in the protected environment of the uterus; a bird's egg requires more protection because it is in a more vulnerable environment, the outdoors.

Answer 3

1. http://edis.ifas.ufl.edu/VM013
"Most good quality eggshells from commercial layers contain approximately 2.2 grams of calcium in the form of calcium carbonate. About 95% of the dry eggshell is calcium carbonate weighing 5.5 grams. The average eggshell contains about .3% phosphorus and .3% magnesium and traces of sodium, potassium, zinc, manganese, iron and copper."

2. Fertilization happens in the human body, not outside.