Abilities that trigger "at the beginning of [your/the] end step" will trigger at the beginning of each of your additional end steps as well.

If a permanent with an ability that triggers "at the beginning of the end step" enters during your first end step, that ability won't trigger during that end step, but it will trigger during your second end step. Likewise, if a delayed triggered ability that triggers "at the beginning of the next end step" is created during your first end step, that ability will trigger during your second end step.

If the target creature is an illegal target as Y'shtola Rhul's ability tries to resolve, it won't resolve and none of its effects will happen. There won't be an additional end step.

Additional end steps don't come with any other additional steps (such as cleanup). Damage marked on permanents won't be removed until you actually get to the cleanup step (which will happen after your second end step), and "until end of turn" and "this turn" effects won't end until the cleanup step either.