[RPG] How long does Alchemist’s Fire burn

Yes, it is consumed.

In the PHB 152 it is described what happens when a part of the floor is drenched with oil and lit (emphasis mine).

You can also pour a flask of oil on the ground to cover a 5-foot-square area, provided that the surface is level. If lit, the oil burns for 2 rounds and deals 5 fire damage to any creature that enters the area or ends its turn in the area.

What it is not told is what happens with the oil on the creature. Logically one may say that the oil would burn for 2 rounds mimicking the mentioned effect. But there are plenty of evidence that fire does not stay in creatures unless specified. For example; Burning hands, Fireball, Delayed Fireball, and Meteor Swarm are spells that clearly specify that no worn objects catch fires, and one of these is a level 9 spell; a mere oil would not light a candle to that. Furthermore, the text ends with a full stop where it should state something in the lines of "every round for two rounds", or so.

It ignites flammable objects in the area that aren't being worn or carried.

If the target takes any fire damage before the oil dries (after 1 minute), the target takes an additional 5 fire damage from the burning oil.

It is important to emphasis that flammable is also a description of oil.

Yes, you get assassinate and sneak attack.

No, the critical would not apply to the fire damage.

Let us assume the DM allows you to craft this contraption, possibly giving it a -2 to hit or something of the sorts due to its unwieldy nature.

PHB, page 96, Sneak Attack:

Once per turn, you can deal an extra 1d6 damage to one creature you hit with an attack if you have advantage on the attack roll. The attack must use a finesse or a ranged weapon.

Sneak attack is applied on any attack roll with advantage as long as it is a ranged or finesse weapon. A bow is a ranged weapon.

PHB, page 97, Assassinate:

You have advantage on attack rolls against any creature that hasn't taken a turn in the combat yet. In addition,any hit you score against a creature that is surprised is a critical hit.

We have our advantage and critical against a surprised creature that hasn't taken their turn yet.

PHB, page 196, Critical Hits:

When you score a critical hit, you get to roll extra dice for the attack's damage against the target. Roll all of the attack's damage dice twice and add them together... If the attack involves other damage dice, such as from the rogue's Sneak Attack feature, you roll those dice twice as well.

The key phrases here is when you score a critical hit. You would roll double the damage dice for the hit on the turn you hit the creature (including double the sneak attack damage), but at start of each of that creature's turns you would roll the 1d4 fire damage separately without doubling it.