MOE1997

They had a visa, but was attempting to use it incorrectly and was turned away and had the visa cancelled. She was attempting to use a B-2 visitor visa to re-enter the country to live. She spent 9 months in the US on the B-2, left for a month, and then tried to be re-admitted, at which point the cancellation and refusal happened.
In contrast, your husband attempted to enter the US but didn't have appropriate documentation, so was advised to get said documentation.

I stopped reading at "I lied".  I have zero sympathy when one misrepresents themselves.

Wow, that is the craziest statement I HAVE EVER READ ON VJ!  Who will support her and her children in Nigeria?  HOW ABOUT YOU???!!!  OMG!!!  You have got to be joking!!!

There is always a chance.
You apply like any other visa.
 

No, the officer allowed OP to voluntarily withdrew his application. He was not deported.

I'm sorry all this happened to you. I think people gave you some good advice. I really wanted to say, hang in there. You are young, it's not the end of the world, though it seems like it now. Most of us here have had or are having long distance relationships, some dragging on for years. Right now you are probably feeling helpless because of what happened. Try to get some plans formed so you can move forward and focus on the future.

No effect, in theory. Though you maybe asked why not continue studies in the US and then you'll have to tell the truth. The answer might give the impression you're visa shopping. Hopefully it will not affect you negatively.
Edit: with UAE passport, you're in better position than some other nationalities but it's still an uphill battle.