How can the fasts of a deceased father be made up, and what is the way to fulfill a forgotten vow?

Good dreams are glad tidings, but legal rulings are not established by them. Regarding the two days your father missed fasting due to his illness: If his illness was incurable, then he owes a fidyah (definition) (expiation) of feeding a poor person for each day (750 grams or, more prudently, 1.5 kilograms), and it is permissible for someone to fast on his behalf. If his illness was curable and he died before being able to make up the fasts, then neither he nor you owe anything. If he was able to make up the fasts but died before doing so, then what was mentioned in the first point applies.

As for the vow you made: If you do not recall having made it originally and relied on the dream, then you owe nothing. If you remember making a vow but do not recall what it was, then you owe an expiation for a broken oath (kaffarat yamin (definition)).