Do an Assassin's Assassinate and a Hunter's Colossus Slayer stack?

Question

Character level 6 - Rogue / Assassin 3 and Ranger / Hunter 3

Assassinate. Starting at 3rd level, you are at your deadliest when you get the drop on your enemies. You have advantage on attack rolls against any creature that has not taken a turn in the combat yet. In addition, any hit you score against a creature that is surprised is a critical hit.

Colossus Slayer. Your tenacity can wear down the most potent foes. When you hit a creature with a weapon attack, the creature takes an extra 1d8 damage if it's below its hit point maximum. You can deal this extra damage only once per turn.

Assuming the attacker has advantage, and the target is a creature that currently has fewer than its maximum hit points, do both skills apply? Or does one or the other apply?

Answer 1

Yes; Unless otherwise noted, different abilities always stack.

5th edition did away with a lot of the specific rules about stacking abilities. Going before another target and the target's possession of less hit points than their maximum are not mutually exclusive so the abilities may stack as long as both conditionals are met.

Does the extra 1d8 get rolled again for a critical hit?

Since you will often be scoring a critical hit with surprise, the question about whether you roll extra damage, like the damage from Colossus Slayer, again may come up. The answer is yes. You can find more details in the question "How does extra damage work for critical hits?"

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What triggers Assassinate?

You seem to be confused about the actual trigger of Assassinate based on the wording of your last paragraph. Advantage is not the trigger, as your last sentence indicates, but instead you simply have to go first (which will give you advantage). If the target is ALSO surprised (see below, emphasis mine) then all of your hits are critical ones.

Surprise

A band of adventurers sneaks up on a bandit camp, springing from the trees to attack them. A gelatinous cube glides down a dungeon passage, unnoticed by the adventurers until the cube engulfs one of them. In these situations, one side of the battle gains surprise over the other.

The GM determines who might be surprised. If neither side tries to be stealthy, they automatically notice each other. Otherwise, the GM compares the Dexterity (Stealth) checks of anyone hiding with the passive Wisdom (Perception) score of each creature on the opposing side. Any character or monster that doesn’t notice a threat is surprised at the start of the encounter.

If you’re surprised, you can’t move or take an action on your first turn of the combat, and you can’t take a reaction until that turn ends. A member of a group can be surprised even if the other members aren’t.