If Trusty Ramhound is defending at a battlefield with an ally, the ally is assigned lethal damage from the combat resolution, and the Trusty Ramhound is assigned two remaining damage, will Trusty Ramhound live?

Yes — Trusty Ramhound lives.

Combat damage is all dealt at once, and the Combat Cleanup kills lethal-damage units before it heals — but the lethal check happens while the ally is still on the board. Ramhound is at 3 Might (2 base + 1 from the ally), so 2 damage isn't lethal; only the ally dies in the kill step, then Combat Cleanup's "heal all units" removes Ramhound's 2 damage before anything can re-check it (Rules 323.5, 466.1.a.1; FAQ #7339/#5961).

A cleaner way to see it: a cleanup can't be interrupted mid-way — nothing between the kill step and the heal step re-evaluates Ramhound's Might. Note the caveat: this survival is specific to combat. If the ally died to spell damage instead, the subsequent non-combat cleanup would have no healing step, and Ramhound would then be killed once its Might dropped to 2 with 2 damage marked (FAQ #7339).