If player A is holding a battlefield with mournful witness, player B moves in with empowered Akali, and deals 2. Player A reacts with Long Sword, player B moves out dealing another 2. Would Mournful witness die before her empower ability triggers? Does combat end before or after the damage is dealt?

Yes — Mournful Witness dies in the cleanup right after the second hit, while the combat is still open, so her "when a combat that I was in ends" trigger never fires; and the damage lands before the combat ends, not after.

Long Sword's Might Bonus (+2, Rule 137.3) makes Mournful Witness a 4-Might unit, so she survives Akali's first move-in trigger (2 damage) and the Combat Showdown actually opens with her as the Defender. When B activates Rogue Assassin during that Showdown, Akali's move-out trigger fires for another 2 — that's 4 damage on a 4-Might unit, which is lethal.

Lethal units are Killed in the cleanup that follows (Rule 323.5), and that cleanup runs while the Showdown/Combat is still ongoing — before the Showdown closes and the combat's resolution step. Combat doesn't end until Step 466.7 (after Combat Cleanup, combat result, and Establish Control), so the damage is dealt first, during the combat, and combat ends afterward. Had she survived to combat end, she'd have been healed by Combat Cleanup and then empowered — but she's already in the trash by then, so no empower.