Is it permissible for each individual family member to give Zakat al-Fitr using a different type of grain, and then for these to be collected and distributed to the poor such that each poor person receives less than a sa' of various types? And is it permissible to purchase the entire quantity and divide the cost equally among family members, despite the differing price of each type, even though each individual has their own income?

There is no impediment to what you have done in giving out Zakat (definition) al-Fitr in the form of food, as long as the food given out is permissible for Zakat, and each of you gives out one sa’. It is permissible to give a portion of the sa’ to one poor person and the remainder to another. Similarly, it is permissible to give several fitras to one needy person, or to distribute a single fitra among more than one needy person. The difference in price does not matter as long as each of you possesses one sa’ of the distributed quantity, with the caveat that Zakat al-Fitr should be given from the staple food that the people of the land subsist on.