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.

At the lethal-damage check during Combat Cleanup (CR 323.4–323.4), your ally is still on the board, so Ramhound's conditional +1 Might is active — it's a 3-Might unit with only 2 damage, which is not lethal. It's not among the units killed in that sweep. Only after the ally dies does Ramhound drop to 2 Might, but by then the single death check has already passed, and Combat Cleanup's next step (CR 468.1.a, "Heal all Units") heals Ramhound's damage before any re-check can occur. This timing is confirmed by FAQ #3588 and #2301: a unit that drops to lethal damage only after another unit dies in the same cleanup survives and is healed.