Does a cavalry unit have to advance to battle again?
Yes. To be eligible for a bonus close combat, a unit (warrior, foot unit with a leader, cavalry, chariot, elephant) must make a successful close combat and advance into the vacated hex.
Does a cavalry unit have to advance and move 1 additional hex to gain a bonus close combat?
No. A cavalry may advance an additional hex before the unit makes a bonus close combat, but it may choose to stay in the first hex it advanced onto and make a bonus close combat. Or it may choose not to make a bonus combat. No combat is forced. A bonus close combat is only possible when the unit advances. So you must advance if you want to do bonus close combat, but you may advance and even move the one additional hex and not do the bonus close combat.
When a cavalry unit has made a successful bonus close combat that eliminates or forces the enemy unit to retreat, can the cavalry advance and move an additional hex?
No. A cavalry unit, after a successful bonus close combat that eliminates or forces the enemy unit to retreat from the hex, may advance onto the vacated hex, but it may not move the additional hex or battle again this turn.