Suppose at time t0 you go short a forward contract with maturity T (and with delivery price equal to the forward price). At time t, t0 < t < T, suppose both the price of the asset and interest rates are unchanged. How much money have you made or lost? (This is sometimes called the carry of the trade.) How does your answer change if the asset pays dividends at constant rate q?

In the case in the question, there is no money made or lost. This is because it was explicitly stated in the question question that both the price of the asset and interest rates are unchanged, and the delivery price equals to forward price.

In a case if the asset pays dividends at constant rate q, the forward price at time t, will be different from the delivery price as there will be change in the price of the assets and interest rates. Thus, money will be gained

Explanation:

The forward price (at time t) of an asset today is the price at which you would agree to buy or sell the asset at the future time, t. The value of a forward contract is zero when you first enter into it. As time passes the underlying asset price changes and the value of the contract may become positive or negative.

there are no gain or loss when the option is contract is exercised early. The question indicates that the asset is an american option. The holder losses funds if the asset pays dividend

Explanation:

when the asset pays dividend it will be wise to exercise the right at an optimal price of the asset.