Yes, it'll deal 4 infect damage. While the creature is already in the graveyard the ability has to look at last known information to determine how much damage is done. LKI says it has infect, so it does 4 infect damage.
Same thing with the second one, except 3 infect damage.
Yes in both cases. That Infect was granted by an instant or an aura makes no difference. The Flailer permanent is the source of the damage dealt by its ability, and since that permanent doesn't exist anymore as the damage is dealt, its last known information is used. It had infect as it last existed, so poison counters it is.
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Given the card in question (Hellhole Flailer) and him not using the card Fling in his question, it's very likely he meant the activated ability on Flailer.
Hellhole Flailer can still deal damage due to its last ability if it was enchanted with Heart of Light, since Heart of Light doesn't change Hellhole Flailer's characteristics or control (it doesn't grant an ability, for example) and will stop applying to Hellhole Flailer once that creature leaves the battlefield (C.R. 611.3a-b, 112.6, 303.4k).
This differs from Phyresis and Bonesplitter in that they both change Hellhole Flailer's characteristics: Phyresis grants an ability to Hellhole Flailer (C.R. 613.1f), and Bonesplitter boosts its power (C.R. 613.1g). And when Hellhole Flailer is sacrificed, its ability takes into account Hellhole Flailer's characteristics at the last moment it was on the battlefield (which could be modified by these two permanents) (C.R. 112.7a).
EDIT (Oct. 3): Added some rule citations.
EDIT (Sep. 28, 2017): One rule was renumbered with Ixalan.
EDIT (Mar. 24, 2018): Clarification.
EDIT (Jul. 24, 2018): Correctness edit.
EDIT (Mar. 25, 2019): Correctness edit.
I fling my enchanted phyresis hellhole flailer at a player. Does that player get poison counters?
Same thing with the second one, except 3 infect damage.
When you cast fling the creature is not the one dealing damage so it won't be modified by any abilities that the creature has.
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if hellhole flailer does deals damage, how is this different than being enchanted with phyresis
lastly, I have hellhole equipped with bonesplitter and use his ability does he do 5 damage? (I didn't unleash him)
This differs from Phyresis and Bonesplitter in that they both change Hellhole Flailer's characteristics: Phyresis grants an ability to Hellhole Flailer (C.R. 613.1f), and Bonesplitter boosts its power (C.R. 613.1g). And when Hellhole Flailer is sacrificed, its ability takes into account Hellhole Flailer's characteristics at the last moment it was on the battlefield (which could be modified by these two permanents) (C.R. 112.7a).
EDIT (Oct. 3): Added some rule citations.
EDIT (Sep. 28, 2017): One rule was renumbered with Ixalan.
EDIT (Mar. 24, 2018): Clarification.
EDIT (Jul. 24, 2018): Correctness edit.
EDIT (Mar. 25, 2019): Correctness edit.