Is it permissible to combine the intention of fasting the Day of Arafah with the intention of making up a missed fast, and will I thereby obtain both rewards?

If you break a voluntary fast, there is no compensatory fast (qada') required of you, according to the majority of scholars, and this is the correct view. However, if the fast was obligatory (such as Ramadan, or its make-up, or a vow, or an expiation), then a compensatory fast is obligatory. It is permissible to combine the intention of making up a missed fast with the intention of fasting the Day of Arafah.