Costs are usually paid before the sale closes. Part of the undertaking which the solicitor gives to the mortgage lender states that the solicitor must be in possession of all outlay necessary to stamp and register both mortgage and property before negotiating the loan cheque.
It would not be normal practice to courier loan cheques unless there was some extreme urgency in closing the sale. Depending on the bank a letter of authorisation may be necessary, so the courier needs to do 2 pick-ups, to solicitor for letter, to bank for cheque, back to solictor with cheque. In City traffic DX can be more reliable and safer. (also cheaper)
I wouldn't defend anyone for not returning phone calls, thats just a matter of good manners but if the solicitor is busy and has nothing to tell you then a call to you may not be a priority. I would still get another member of staff to return the call as a matter of courtesy but not everyone thinks this way