Which electron shell determines the chemical properties of an atom?
Primarily the outermost shell.
What is an electron that is found in the outermost shell of an atom and determines the atom's chemical properties?
It is called a valence shell electron. The number of valence shell electrons atoms of an element have can be read off the periodic table: Those in group one have one valence electron, those in group two have two etc. Thus, elements in the same group have the same number of valence shell electrons and so, similar chemical properties.
Different elements have different chemical properties because they have different atomic structures and characteristics. Different chemical elements have different atomic structures, and particularly different electron shell structures. Chemical bonding is generally based on electron structures of atoms, and as these structures vary (or are similar) so the chemical properties vary (or are similar).
Why do members of the same vertical group on the periodic chart tend to show similar chemical and physical property?
Does the number of electrons in the outermost energy level determine the chemical properites of an atom?
The noble gases (helium, neon, krypton, xenon, radon) have complete outer electron shells, and the halogens (fluorine, chlorine, bromine, iodine, astatine) have almost complete outer electron shells. Note that the chemical properties of an element with a complete outer shell are tremendously different from the chemical properties of an element with an almost complete outer shell.