There is no restriction in law preventing a non-UK resident person being appointed as a director of a UK registered company. Generally there is no requirement for a director to live in the UK either.
As long as articles of association and memorandum of the UK company do not restrict the appointment of non-UK resident persons as directors and the person is not disqualified themselves from being a director then there no reason why such a person could not be appointed as a director.
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This is probably a separate question, but this would be dealt with in the company accounts of the UK registered company. Profits would be allocated in accordance company policy as voted by the directors of the company. Directors may be paid salaries and the company may pay dividends to the shareholders of the company or choose to put the profits back in to the business.
All of these transactions would be detailed in the company's business accounts
Similarly the financing (loans/equity investment) of the company would also be included in the company's UK accounts.
These accounts (eg. profit and loss accounts, and balance sheet) must be submitted to Companies House each year. The Companies House website contains useful information on the compliance issues for UK companies:-
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