What is the ruling on giving money to a needy friend for a project on the condition that he returns it with a share of the profits? And is it necessary to write a receipt for the amount despite the trust between the two parties?

The described scenario is Islamically forbidden, and a partnership with a capital guarantee is invalid. Losses in partnerships are borne by all partners according to each one's capital share, and any condition to the contrary is void. This transaction is a loan with interest for the lender, and this is usury (riba). Jurists have reached a consensus that it is forbidden for a creditor to benefit from the act of indebtedness if it is stipulated as a condition, and every loan that yields a benefit is usury.

The permissible approach is for losses to be borne by you proportionate to your capital, and your share of the profit should be a common percentage thereof. As for documenting debts in writing, it is recommended but not obligatory.