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 survives.

Here's why: Trusty Ramhound is Might 3 while its ally is present (base 2 + the +1 Might from its ability). At the combat cleanup's kill check, the game evaluates lethal damage for all units at one point in time while the ally is still alive — so Ramhound has only 2 damage marked against Might 3, which is not lethal. The ally is killed in that same simultaneous event, and then the Combat Special Cleanup's "Heal all Units" step (Rule 466.1.a.1) removes Ramhound's 2 damage before any later check can kill it. Even though Ramhound's Might drops to 2 when the ally dies, its damage is already healed.

This matches the established Lee Sin aura rulings (FAQ #5961, #7339) and FAQ #2301: "A unit that would drop to lethal damage after another unit dies will survive if it heals during the cleanup." Note that this only works because healing happens during the same combat cleanup — outside combat (e.g., a kill spell removing the ally), Ramhound would take lethal in the subsequent cleanup.