I got your order and have processed it
This is okay; there's an action in the past, and a just-completed action, presented in the correct order.
She has agreed to my terms, so I only did my part yesterday.
I don't like this one; it's kind of a "garden-path" sentence, where when you start reading it you automatically assume one thing and when you're done you have to go back and think "oh yeah, I suppose they could have meant it that way." The default reading is that "she has agreed" is a just-completed action; then as a consequence of that, you cannot have an action entirely in the past. You then have to think "oh, they must have meant 'has agreed' as an ongoing thing that started at some point in the past (prior to yesterday)." So while it is perhaps technically justifiable, it's still misleading.
[She] quietly slipped away today from the hospital where she has been staying
This one is okay; you can "slip away" from someplace that you're staying as long as you return there. She might have slipped away to go visit a friend, or to go to a birthday party, or something.
If you have just registered, you received your current bill via USPS.
This is a conditional statement with a lot of implied bits to it, but I believe it is okay. The things going into this are:
- we normally bill via USPS
- when you register, you can change how you receive your bill
- you weren't registered before
- in order to get to this point, you must have been billed at least once already
- therefore, since you are at this point, it must be the case that your most recent bill (that is, the "current" one) was sent by USPS.
This part on the invoice signifies the quantity shipped but that part signifies the quantity he had wanted.
I think this would be better with the simple past, but someone could make a case for the past perfect, since the sequence of events is that at time A in the past, someone wanted X units; later (but still in the past), the company shipped Y units; and now, we are talking about it.
There had been an error on the exam, so here are the correct answers.
This one is clearly wrong, since there is no "later, but still in the past" event to make the past perfect useful. (It should be "There was an error.")