Time Complexity and graphs
Yes, anything that has constant input could be done in $O(1)$. Howether if the coordinates have size more than $O(1)$ than you will not be able to even read them in time.
That depends on your representation of the graph and what operations do you count. If you assume that you read the whole graph in some way and then find the needed degree, the complexity will be $O(|V| + |E|)$ if your representation is an edge list, $O(|V|^2)$ if it is an adjacency matrix. If you assume that you already have some representation of the graph in memory you can find the degree in $O(|V|)$ if you have the adjacency matrix. 
You can do it in $O(|V| + |E|)$, start with an empty graph (all the degrees are zero) and add the edges one by one. When you add an edge to the graph only two vertices change their degrees.
No, consider the graph $K_4$ (complete graph on $4$ vertices). All the degrees in it are odd so it does not have even en Euler path.
If you want to read the coordinates it takes $O(n)$ operations, so no.