Let us say that graph $G$ is quasi-connected iff for every two vertices there is either a finite path or an infinite path in sense of the offtopic of this answer.
Assuming the axiom of choice, is it true that every quasi-connected graph has a "spanning tree" (that is a subgraph where every pair of vertices is connected by unique finite or infinite path)?
Best Answer
Yes.
First choose an infinite ray in each of the (ordinarily) connected components, and extend it to a spanning tree.
Then, as long as there is still a subgraph isomorphic to $\mathbb Z$ anywhere in the graph, select one of its edges and remove it. At some point (after possibly transfinitely many edge removals) you cannot remove more edges, and the graph is now a forest of trees with exactly one infinite branch each.
And a "tree" in the sense you define in your final parenthesis is exactly either an ordinary tree, or a forest of ordinary trees that each have exactly one infinite branch. To connect two nodes in different components you need to go from each of the nodes out the infinite branch of its component, and then connect those two rays at infinity.