[RPG] If you take action to automatically end persistent damage, do you still take the damage that turn

They do not stack because the damage roll of Flames of Phlegethos is separate from any spell's damage roll

The first bullet of Flames of Phlegethos states:

When you roll fire damage for a spell you cast, you can reroll any roll of 1 on the fire damage dice and use the new roll.

And thus it can only apply to damage rolls of spells you cast. The other bullet however, states:

Whenever you cast a spell that deals fire damage you can cause flames to wreathe around you until the end of your next turn. .. any creature within 5 feet of you that hits you with a melee attack takes 1d4 fire damage.

The damage from this is not you casting a spell, it simply happens when you are hit by a melee attack. Even though the spell "causes flames to wreathe around you" , these flames are not considered part of the spell as they are not in its description. This let's us establish that the damage the flames cause is not part of the spell's damage roll. Thus the parts of this feat cannot be used together because the additional 1d4 will always be its own damage roll, separate from any spell's.

NO.

The key phrase here is “persistent damage equal to and of the same type as the bomb’s splash damage.”

So we’re doing an amount equal to our splash damage. Now, what are the rules for critical hits and Splash damage?

Most bombs also have the splash trait. When you use a thrown weapon with the splash trait, you don't add your Strength modifier to the damage roll. If an attack with a splash weapon fails, succeeds, or critically succeeds, all creatures within 5 feet of the target (including the target) take the listed splash damage. On a critical failure, the bomb misses entirely, dealing no damage. Add splash damage together with the initial damage against the target before applying the target's resistance or weakness. You don't multiply splash damage on a critical hit.

Since the splash damage is not multiplied on a critical hit, the persistent damage, which is equal to that splash damage, is also not multiplied.

To use the example from the linked text on Nethys;

For example, if you throw a lesser acid flask and hit your target, that creature takes 1 acid damage, 1d6 persistent acid damage, and 1 acid splash damage. All other creatures within 5 feet of it take 1 acid splash damage. On a critical hit, the target takes 2 acid damage and 2d6 persistent acid damage, but the splash damage is still 1. If you miss, the target and all creatures within 5 feet take only 1 splash damage. If you critically fail, no one takes any damage.

With Sticky Bombs added to this equation, on either a normal or a critical hit, you would add the 1 point of splash damage, as persistent damage, to the 1 or 2d6 persistent acid damage (on a hit or crit respectively) caused by the base bomb. These numbers are combined into a single persistent damage effect.