a) number it the other direction. You named it correctly (2-pentene), but you drew it with the double bond at position 3. The substituents should be carbons 1 and 3, not 2 and 5.
b) Zaitsev's rule follows from the mechanism. In the transition state (at least for E2), the base has partially removed the beta proton (the proton on the carbon adjacent to the carbon bearing the leaving group), the electrons from the carbon-hydrogen bond are partially collapsing to form the new pi bond, and the leaving group has partially left. Double bond character is so highly developed in this transition state that the relative stability of the alkenes determines which alkene is the major product. (Same is true for E1, but the leaving group has already left...)
Put another way, more substituted alkenes are more stable than less substituted alkenes. The transition state energy leading to more substituted alkenes is lower. This means the activation energy leading to the transition state leading to the more substituted alkene is lower. This means the more substituted alkene is formed preferentially over the less substituted alkene.