I do not understand what 'everything lies on the plane' means. I take it as meaning 'everything is laid out in two dimensions' but that doesn't make sense. Can HN help?
Nelson asked: What is the smallest number of colors that you’d need to color any such graph, even one formed by linking an infinite number of vertices?
The Wikipedia page describes the infinite-vertices version of this graph as
an infinite graph with all points of the plane as vertices and with an edge between two vertices if and only if the distance between the two points is 1.
This of course is impossible to draw but Wikipedia shows seven-vertex and ten-vertex subgraphs of it:
https://en.wikipedia.org/wiki/Hadwiger%E2%80%93Nelson_proble...