Only if you share a common ancestor
Probably not. You are technically only family if you share a common ancestor. If you do share a common ancestor, then they will be your 2nd cousin as well. It is not likely (but possible) that you are related to them.
You are related to some of your cousin's cousins, and not to others. If you have an ancestor in common, you are related. If you do not can a common ancestor somewhere, then you are not related.
There is probably no relationship at all. You must share a common ancestor to be related. A cousin's cousin is not likely to share anyone, but it is certainly possible.
The parents of your second cousins are your second cousins once removed, since the ancestor that you have in common with then is your great grandparent.
You do share a common ancestor. You parent would be a fifth cousin one removed. You would be a fifth cousin twice removed.
They are out of synch to a common ancestor by one generation. That would make the first cousins once removed.
They would be second cousins, once removed. The descent is from a common ancestor. Thus 1st cousins share a grandparent as a common ancestor. 2nd cousins share a great-grandparent as an ancestor. If the descent from the common ancestor is not an equal number of generations, the inequality is expressed as 'removed' from the oldest generation being compared.
It will depend on whether you share a common ancestor (is the niece's parent also a cousin?). If you do not, you are technically not related. If you do (you both have the same grandparents) you would be second cousins.
If two related people are described as "Cousins, once removed" it means that one of them is one generation further removed from their youngest common ancestor than is the other. If Joe and Jane are First Cousins, once removed, and their youngest common ancestor is Joe's grandfather, then that common ancestor is Jane's great grandfather. If Joe and Jane are Second Cousins, once removed, and their youngest common ancestor is Joe's great grandfather, then that common ancestor is Jane's great great grandfather - and Jane's parent is Joe's Second Cousin.
Yes, Fifth Cousins are descended from a common great great great great grandparent. All cousins are blood related because there is a common ancestor. The only reason any cousin would not be blood related is if there was an adoption.
You need to find out if you have a common ancestor, as this is what determines if you are cousins. If your common ancestor is your grandfather, you are 1st cousins. If your common ancestor is your great-grandfather, you are 2nd cousins. If your common ancestor is your grandfather and their great-grandfather, you are 1st cousins once removed. It can get complicated, so once you find your common ancestor, come back and let us know, and someone can tell you exactly.
First cousins' grandchildren are third cousins to each other.