My question is
Let $G$ be an infinite, connected, locally finite, vertex-transitive graph. Must $G$ have the following substructures?
i) a leafless spanning tree;
ii) a spanning forest consisting of rays;
iii) a spanning forest consisting of double-rays.
A ray is a 1-way infinite (self-avoiding) path. A double-ray is a 2-way infinite path. A tree $T$ is leafless, if it has no vertex of degree 1 (a leaf). Equivalently, every edge of $T$ lies in a double-ray.
The following implications hold for every (infinite, connected, locally finite) graph G, vertex-transitive or not:
iii) trivially implies ii), by removing one edge from each double-ray.
iii) also implies i): given a spanning forest $F$ of double-rays, let $T$ be a minimal tree containing $F$, which can be constructed greedily by connecting the components of $F$ one by one, thereby never creating a leaf.
i) implies ii): pick a root $r$ of the leafless tree $T$, and a ray $R$ containing $r$. Delete all edges of $T$ incident with $R$ but not contained in it. Repeat in each resulting component, picking the vertex adjacent with $R$ as the root.
The other implications can fail for general $G$, but I don't know if they hold when $G$ is vertex-transitive.
Note that if $G$ is a Cayley-graph, and one of its generators has infinite order, then iii) holds. In 2009 I asked if every 1-ended, finitely generated, (connected) Cayley graph has a spanning double ray (Problem 3 in Infinite Hamilton cycles in squares of locally finite graphs), which is much stronger than iii) above, and still open as far as I know. It follows from the main result of the same paper that for every vertex-transitive $G$, the square $G^2$ satisfies iii). Here, $G^2$ is defined by adding an edge between each two vertices that are at distance 2 in $G$. In particular, every f.g. group has a f.g. Cayley graph satisfying iii) (and hence i) and ii)).