Doubts regarding the validity of wudhu when one has an oily face

Q: My face is oily. When I am performing wudhu, some of the oil of my face gets onto my hands. Do I have to rub my hands with water to ensure that water gets underneath the oil and reach my skin?

Do I have to rub any part of my body if the oil of my face gets onto them (for example my forearms, my feet)?

In that same logic, if oil from a part of my face (the left part for example) get onto or mixes with the oil of another part of my face (the right part for example) do I have to rub that area with water (the right part of my face)? 

Giving zakaat to a person that one thought was eligible to accept zakaat

Q: I would like some advice on the following matter:

During this year's Ramadhaan, I distributed a portion of my zakaat to a person who was previously regarded as eligible for receiving zakaat. Today, I found out from a family member that after enquiring from their local Ulama, they found out that this person is no longer a recipient of zakaat.

Please advise whether the zakaat will be accepted or whether I have to distribute the sum of that amount again as Zakaat.