BP, in their own website, states "We are the largest non-US company on the New York Stock Exchange." Their head office, and legal operations, are handled from there.
Stock ownership does not determine the ownership of a company, unless it has control of the Board and legal operations.

All this posted above.

Stock Ownership 44% UK, 39% US; Institutional ownership 33% UK, 25% US, rest individuals.
Also posted above.

Stock ownership may be changing rapidly as many are divesting their portfolios of BP stock and reflected by the current low stock value.
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The well has drilled into sediments below thick salt layers and rock which formed an effective seal, trapping the oil and gas, and leading to high gas pressure, leading to the danger of blowouts.
The mantle is not involved; at this location it is much deeper.
Temperatures are high, however, also leading to high gas pressures.

Several companies have drilled test wells into these sediments below the salt layers; it is risky because if something goes wrong or there is carelessness, blowout like BP's may occur and corrective measures are experimental or poorly tested.
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BP used to be called British Petroleum; as they increasingly operated all over the world, the Board decided the name should be simply BP. Of course many people still call the company 'British Petroleum'. So what? Not important.