To the first question, yes. This is a common structure for creating subsidiaries, operating units, and joint ventures. If the LLC just has a single member--the C Corp parent--then it will be disregarded for tax purposes, and all LLC revenue and expenses will be reflected on the parent's tax return. If the LLC has multiple members--as in the case of a joint venture--the LLC's revenue and expenses will be allocated as set out in the operating agreement. (Yes, you need an operating agreement. And as with any multi-member LLC, the taxes can get pretty complicated.) As for funding, in each scenario that simply involves moving money from the C Corp to the LLC, either in exchange for membership interests or as a subsequent capital contribution. Money _can_ move back and forth between the entities, just make sure you're keeping records and obeying tax laws.
To the second question, strictly speaking the answer is no, the LLC can't have the same directors because LLCs do not have directors. LLCs have "members," which are the LLC equivalent of stockholders, and "managers," who oversee operations. Managers are the closest thing an LLC has to a directors, and in some cases managers are organized as a board. But managers often have more day-to-day responsibility than corporate directors. Conceptually they are somewhat of a director/officer hybrid. An LLC can also have true officers, though LLC statutes give them little or no attention, so you'll have to define all officer roles and responsibilities in the operating agreement.
The directors of the C Corp parent can absolutely serve as the managers of an LLC subsidiary, whether wholly owned or a joint venture. Keep an eye out for fiduciary conflicts in these cases. They can be waived pretty easily in the LLC operating agreement--which is common in joint ventures--but not so with the corporation. You might also have a C Corp officer serve as the manager of an LLC if the LLC holds a subsidiary in that officer's domain.
A word of warning. Increasingly courts are holding that an LLC structured to mimic a corporation--particularly one with a board of managers that looks like a corporate board of directors--will be subject to corporate-style fiduciary duties, including obligations to members equivalent to the obligations a board owes to its shareholders in the corporate context. This can be a serious problem in many situations, particularly joint ventures where each venturer has an independent, often competing operation.