Cooking frozen food after it has defrosted. Is the remaining food ok to go back in the fridge?

If they were thoroughly cooked, then they're probably okay. The problem is that uncooked, unrefrigerated (or unfrozen) food is a playground for bacteria. While cooking the food will usually eradicate the bacteria, some forms of bacteria essentially go into protective hibernation and will re-emerge once the food's temperature drops; it's therefore best to eat the food right away and let your immune system get the critters before they can re-emerge.