From the FAQ:
Q: Can I trigger the Action on a Capital Center if there are no tokens on the card?
A: If there is no token to remove, you cannot gain the benefit of the Capital Center. The word “Then” means that the previous ability in the effect must have resolved successfully before the “Then” clause is resolved itself. So you can trigger the Action, but it will have no effect.
For the second question, you can do it once, since it's a triggered action and you can only activate it once per trigger condition, in this case, Beginning the turn. If you could Begin the turn several times per turn, you could trigger it several times, but that's not the case:
Trigger Conditions (v1.4)
A trigger condition is anything that must happen before a certain card ability can be triggered. This could be playing a card, using another card ability, or even a Constant Effect.
For example: The trigger condition on Walking Sacrifice (AU 23) is “When this unit leaves play...” for the unit must leave play before the Forced Effect can trigger. Another example of a trigger condition is “At the beginning of your turn...” which is found on cards such as Rat Ogres (CC 55) and Cloud of Flies (CS 93)
Triggered Actions (formerly Response Actions) (v1.4)
Triggered Actions are a subset of Actions that contain a trigger condition. Each Triggered Action can be triggered once per copy of the Triggered Action and only once per trigger condition. If this trigger condition is met during the resolution of other effects (or outside of an Action Window) then the Triggered Action must be played during the first available opportunity once an Action Window opens, or it cannot be played at all.
For example: Rat Ogres (CC 55) reads “Action: At the beginning of your turn, restore all Skaven units.” The trigger is at the beginning of your turn, and the action can only be triggered once per copy of the card in play.