Is it obligatory to calculate the saved money with the business money to pay Zakat on both together, or to pay Zakat on each separately, and is the saved money that is added to the business money for a few months and then reclaimed considered a debt owed by the two partners?

As for the saved money that does not reach the nisab (the minimum amount liable for zakat) except by combining it with your share in the company, its hawl (the one-year period for zakat calculation) will be unified with the company's money. It begins from the day you possess the nisab, and you pay zakat on all of it when its hawl is due. However, if the saved money reaches the nisab on its own, it has an independent hawl.

Regarding money placed in the company as a loan: half of it has not left your possession, so you pay zakat on it along with your own money. The other half is a loan to your brother; you also pay zakat on it as long as the debtor is solvent and willing to pay. Otherwise, there is no zakat on it until you receive it. If the amount is a new participation in the company, you pay zakat on this amount and its profits along with the hawl of the saved money.