"the sense" actually goes back to "explain." The model explains 2 things: 1. how so much of what we do can be emotionally driven 2. the sense in which the emotions have their own reasons and their own logic so the question becomes: What does the model explain? The sense in which emotions have their own reasons

It means: There are very few signs of recovery. The signs that do exist are not that strong.

The grammatically correct answer is technically "Every student has his or her own task given by the teacher," but as mentioned, when spoken, you'll often hear people use "their": Every student has their task set by their teacher. So the first way is the technically correct way, but the second way is more often used in informal situations.

Any mistakes in this email?

Hi Neil,
Regarding the photos you sent yesterday, our technician informed us that it was the drive connector not the motor connector.
The motor has two wires: One is the power wire and the other one is the drive wire.
We need photos of these two wires, so please have your technician contact us.
We will be awaiting your reply.

Re: Response to Warning Letter.

Dear Mr. Takagi, 

Regarding the material shipment timetable, we checked with our supplier about the shipment's progress. We made sure that most of the material collected and the set of materials will be ready in one week. After that we will provide you with the exact delivery date to Djibouti. Additionally, for the pump, Pedrollo delivery is planned in 5 weeks to the supplier warehouse, so we told the supplier that the pumps should be delivered in partial shipments.
We have attached the table below to show you the status of the materials and if they are ready for delivery or not.
2- We have attached for you the general schedule for the project.
3- We have attached for you the details of the requested material with specifications and quantities. For more details about the specifications, please refer to the catalog that we provided previously, as nothing has changed.

Sincerely,