Missatges enviats per Tim Hunt

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If you are in a public function, then $this should be defined. (Unless something is doing some evil low-level hackers to call this method in a non-standard way.)

Anyway, this error does not happen in the standard Moodle release. (This functionality is covered by automates tests which pass.) So, that suggest the problem is caused by whatever customisation you have done in your version of Moodle. I hope you can work it out.
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I don't entirely understand either, but approximately:

The gradebook has a column for each graded activity. So, when you add an activity, the gradebook structure has changed, which may affect totals and things, and so Moodle had to recalculate the totals for each student, and that takes time sometimes - so there is a progress bar so you know what is happening.
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Ordering is tricky, because the student answers in-place. There is no way to distinguish that the student intentionally sumitted an answer identical to the orgininal random order, compared to them not intending to submit anything. So, I think the current behaviour is OK.

The essay issue is just a bug, I think. Probably what is happening is that when the HTML editor is blank, then some HTML is being sent back (like <p></p>) and Moodle is incorrectly considering that to be an answer. We should probalby be using Moodle's html_is_blank function here. So, please make a Tracker issue for this.