Should one who finds a lost item while working on his partner's car share this lost item with his partner, or does he take it alone, with the partner's knowledge of it?

The found item belongs to the one who found it, and the partner has no right to it. If sharing it were stipulated in the partnership, the partnership would be invalid due to the uncertainty (gharar) involved and the guarantee of something not required by the contract. Thus, whoever takes the found item has the most right to it, and he is not obligated to share it with his partner.