If a creature in Magic the Gathering is regenerated does its kickers stay on it?

When a creature regenerates, it ignores a destruction effect, or destruction from lethal damage, it removes all damage from itself, and it becomes tapped.

There is no other change. Auras and Equips will remain attached, any counters or conditions on that creature will remain, etc. If the creature gained any bonuses from being kicked, then those will remain too.

Think of regeneration as the "Gandalf Bubble". It protects you from damage (the damage never technically reaches through the bubble). Other than tapping you and stopping the damage, it will have absolutely no affect.

^I see where you're going with it, but not an accurate analogy. The damage is still 'dealt', ie, goes through the 'bubble', the damage is not prevented, just removed, as well as the 'lethal damage destruction' state-based effect being ignored. So if anything triggers from or is applied due to the damage being dealt (like Lifelink) then that all still happens.