Yes.
If the grandfathers were 1st cousins, the grandchildren of each grandfather are 3rd cousins to each other.
If your great grandfather's were brothers, then you are third cousins.
Two people whose grandfathers were brothers are second cousins to each other.
Your grandfather's brothers would be the uncles of your parent. That would make them your great uncle.
Since your great grandfather was brother to the other person's great grandfather, the two great grandfathers had at least one parent in common. That parent is your common ancestor, the great great grandparent of both of you. People who share a grandparent are first cousins. People who share a great grandparent are second cousins. People who share a great great grandparent are third cousins. You are third cousins.
It would technically be considered your half- cousin since you do have the same grandmother but a different grandfather.
Your grandfathers' sisters' grandchildren are your second cousins.
3rd cousins
Since your great grandfather was brother to the other person's great grandfather, the two great grandfathers had at least one parent in common. That parent is your common ancestor, the great great grandparent of both of you. People who share a grandparent are first cousins. People who share a great grandparent are second cousins. People who share a great great grandparent are third cousins. You are third cousins.
You would be second cousins.
The relationship to the grandfather or the relationship to the grandson of the grandfather? Let's say your name is Ken, your grandfather's name is William, his brother's name is Eli, and Eli's son is named John. If you're asking about William and John's relationship, they are first cousins. If you're asking about Ken and John's relationship, they are first cousins once-removed.
My Paternal grandfather and his sister were twins. I myself have fraternal twin boys. But none of my 1st cousins, or my sister have twins. I have two cousins who have had 3 single births so far, and my sister has had 2 single births as well. I dont know the chances of them giving birth to twins, but it does seem relatively low.