How to delete a scheduled task using Rules Scheduler?
Have a look at my answer to "Rule to store user&#39;s last post date and scheduled rule based 30 day reminder" for an illustration about how to delete scheduled tasks, using the "schedule_delete". As explain in that answer: the schedule_delete that is part of this rule is to ensure that previously "scheduled" reminders will get removed when updating the data field (so that it only triggers 1 reminder, and with the most recent date).

To make this work (that you actually indicate the exact scheduled task to be deleted), you need to specify an appropriate "identifier", like "[account:name]" in my example. To understand how I know what in my case the identifier is of the task to be deleted, look at the way I specified my (new) scheduled task to be created: it is "identifier" : "[account:name]", which  is what I used as my identifier to schedule a task.

In your case, I think "[account:name]" would be a pretty good choice for such identifier. But your questions states "... users then gets 4 emails over the next four weeks, sent by the scheduler ...". So if the 2nd email only gets scheduled after the first email (and not schedule all 4 emails upfront together), and similar for the 3rd and 4th email, then you should be fine using this identifier also.

Note: Specifying an "identifier" like "[account:name]" is what you'd do in your Rules Action (similar to my "step 3" in the above sample). While you'd probably want to create a Rules component similar to my "step 2" in the above sample to actually send an eMail.

Update: The "integrity check" problem shown near the end of the updated question, is probably because in your rule, as shown in the updated question, does not specify the date to schedule your component. Have a look again at my answer I mentioned above: the rule there specifies a date via this part of it:

  "date" : {
    "select" : "account:field-next-article",
    "date_offset" : { "value" : 2592000 }
  },