If it is not your common grandparent, there is no relationship.
You and your first cousins share the same Grandmother.
The children of your grandmother's first cousin are your second cousins, once removed.
The children of your grandmother's brother (of your great uncle) are your parents first cousins and your first cousins, once removed. The grandchildren of your great uncle are your second cousins.
If you are referring to YOUR grandmother and grandfather then your children are their great grandchildren. If you are referring to your parents, then your children are their grandchildren. The primary kinship dominates.
If your grandmother and his grandmother were first cousins, then you and he share a great grandparent and are third cousins. You can date anyone your want to, and a third cousin, or even a first cousin, is not exception to that. Some people get concerned when cousins become sexually active with each other, but very few would be concerned about a marriage between third cousins, let alone something less obvious.
You are First Cousins
Your grandson and your sister's grandson are second cousins to each other.
Yes. If your grandmother's first cousin was someone else's grandmother, then the two of you are third cousins to each other.
Your first cousin's grandchild is your first cousin, twice removed. Your second cousin's grandchild is your second cousins, twice removed. Your third cousin's grandchild is your third cousin, twice removed.
The grandson of your grandmother's first cousin is your third cousin.The grandson of your grandmother's second cousin is your fourth cousin.The grandson of your grandmother's third cousin is your fifth cousin.Other possibilities arise if your grandmother's "cousin" is once removed, twice removed, etc.
Your fiancee's father and your grandmother were brother and sister. You fiance is first cousin with one of your parents. So you would be first cousins once removed.
First cousins