Yes, absolutely - although this may not be the best option from the confidentiality point of view.
Yes, corporate shareholders are allowed. Corporate directors, too.
A nominee director, also known under the more modern name of "third-party director", is a third party, who would be officially registered and acts as a Director of a company. The main objective of having a nominee director is to prevent the general public from knowing that a direct relationship actually exists between the beneficial owner and his company. Employing a nominee director or a nominee manager for an offshore company helps to offset potential implications that the beneficial owner is closely involved in the control and operation of the particular company (the "management and control" concept). So, in essence, the main purpose of the nominee director is to facilitate the confidentiality of the client. The nominee director may perform various administrative functions which otherwise would have to be performed by somebody directly related to the company. The actual involvement of the nominee Director in the day-to-day matters of the company may vary in accordance with the necessities, wishes, and circumstances of the client. Quite often (but not always), the involvement of the nominee remains, true to the name, nominal. In such case all practical functions of the day-to-day management of the company would be routinely carried out by the client himself, acting as a "representative" of the company on grounds of a power of attorney. Nominee directors are provided by to their client companies as an optional service.

A nominee shareholder is an unrelated third party, who is officially registered as the holder of shares in a company. The purpose of the nominee shareholder is to uphold client secrecy by shielding the actual owner of the company from being publicly associated with that particular company. In a nominee shareholder situation, a confidential legal document (a declaration of trust, a deed of transfer, a nominee services agreement or another similar document) would be issued by the nominee and held by the beneficial owner. Such confidential document would evidence the actual state of affairs - namely, that the shares are only held by the nominee for and on behalf of the beneficial owner, and that only the beneficial owner has the right to dispose of those shares and is entitled to all benefits and profits deriving from those shares.

Nominee shareholdings are commonly provided by to their client companies as an optional service.