All of that looks like sloppiness, however your final point about them sending the money to the wrong person is the key issue, all barclays accounts show as Leicester so that is fine, but it sounds like a digit or digits have been entered incorrectly, the issue being did you provide incorrect details or did they input incorrectly, unfortunately as you provided verbally it will be difficult to prove either way. Lexus will have to try to get the payment returned, which could take some time, they may feel they have paid the refund in good faith, so may be reluctant to pay you until they get their money back.
I know exactly how you feel about the car, I had a relatively minor issue with mine when I bought it, and the whole thing really took the shine off the car, and at the time I just wanted to get rid of it, now the issue is resolved I feel better about the car, but the"magic"is still not there, hopefully that will improve with time.