Will the ownership of the company's shares transfer to me if my father nominally registered them in my name due to debts, and if not, should I accede to my father's request to sign forbidden or doubtful contracts that he would have signed himself if the shares were in his name? And is it considered valid to purchase machinery from a manager of a foreign company who holds a legally sound power of attorney, even if there is a possibility of a legal dispute between him and the foreign partners, despite it being merely a suspicion?

If your father transferred shares to your name to evade repaying his outstanding debts, then this is not permissible for him. He is obligated to promptly settle his debts, due to what is established in the Sunnah regarding the prohibition of a wealthy person procrastinating on debt repayment. This nominal sale (a sale of necessity, or a "talji'ah" sale) is invalid according to the Hanafis and Hanbalis.

Therefore, this sale is not valid, and it does not entail any effects, such as the ownership of the shares. It is not permissible for you to sign forbidden contracts. You must advise your father gently and kindly not to delay in settling his debts. As for purchasing machinery from a company owned by foreigners, it is permissible unless you know that the manager is unauthorized to act, or that the partners refuse the sale, because the default assumption in the affairs of Muslims is validity.