Does the customer get emails while their return is being processed?

Yes. With the Return Settings plugin, Wcart emails the customer at each step:

• When they submit a return request
• When you click Approve or Reject on Orders → Returns
• When you click Complete on an approved return, telling them it's refunded or exchanged

Each email is listed in the order's Timeline, for example "Return approved email sent to …", so you can check it went out.

The emails go through your store's email provider, so make sure Settings → Web Settings → Email Provider is set up.