Yes.

You're assuming more than what's necessary.

For an isometric group action on a Gromov-hyperbolic space, you have 5 possibilities (Gromov's classification):

• (a) bounded orbits
• (b) horocyclic (fixes a unique point at infinity, no loxodromic element; preserves "horospheres")
• (c) axial (preserves an axis, on which some element acts loxodromically)
• (d) focal (fixes a unique point at infinity, existence of a loxodromic element)
• (e) general type (= other): implies the existence of a non-abelian free subgroup acting metrically properly.

(a), (b), (c) are clearly ruled out in your setting (transitivity, and valency; valency $\ge 3$ would be enough). (d) is ruled out because discrete groups have no metrically proper focal action at all. So (e) holds.

The case of actions on trees is an useful motivating baby case in the above "classification"; all cases can already occur.

Yes.

You're assuming more than what's necessary.

For an isometric group action on a Gromov-hyperbolic space, you have 5 possibilities (Gromov's classification):

• (a) bounded orbits
• (b) horocyclic (fixes a unique point at infinity, no loxodromic element; preserves "horospheres")
• (c) axial (preserves an axis, on which some element acts loxodromically)
• (d) focal (fixes a unique point at infinity, existence of a loxodromic element)
• (e) general type (= other): implies the existence of a non-abelian free subgroup acting metrically properly.

(a), (b), (c) are clearly ruled out in your setting (transitivity, and valency; valency $\ge 3$ would be enough). (d) is ruled out because discrete groups have no metrically proper focal action at all. So (e) holds.

The case of actions on trees is a useful motivating baby case illustrating the above "classification"; all cases can actually occur.