So if lethal damage can't kill an indestructible creature, what do we do about the damage itself? Is it acknowledged? Say you have an effect that doesnt let a creature block if it has taken damage this turn, can that creature with indestructible block?
Creatures with indestructible can take damage like normal, and will still be counted as having that damage for anything that cares about that.
The full definition of Indestructible is:
702.12a Indestructible is a static ability.
702.12b A permanent with indestructible can't be destroyed. Such permanents aren't destroyed by lethal damage, and they ignore the state-based action that checks for lethal damage (see rule 704.5g).
It does not prevent damage, so it takes damage like normal.
Absolutely. Indestructible does not prevent damage, it prevents damage from causing a creature to be destroyed. That damage needs to be done for lifelink to happen, and in the case of wither or infect, still will kill an indestructible creature due to loss of toughness. The damage is dealt, and once dealt it is handled like all other damage, with the exception that otherwise lethal damage can't destroy the creature.
Consider cards like Bonds of Mortality or Burn from Within, cards that remove indestructible. You could activate Bonds of Mortality or cast Burn from Within after the creature has already been dealt lethal damage, causing it to lose indestructible and die. Burn from Within as written would be useless if damage was untracked due to indestructible, since the damage done by the spell would not be counted, and the second half of the ability checks if the creature was dealt damage by the first half.
Yes. If an indestructible creature takes "lethal" damage (under the normal definition of "lethal") and then loses Indestructible at a later point in the turn, it will still be destroyed.