response from our hosting company. I have no idea what it means or if they are correct. Can someone please help.

Checking the access logs for "dropship" there are 302 "file not found" errors (full log can be downloaded from cPanel -> Logs -> Raw Access Log)

86.151.8.123 - - [02/Jul/2013:17:21:20 +0100] "GET /index.php?main_page=checkout%5fprocess&referer=paypal HTTP/1.0" 302 - "-" "Mozilla/5.0 (Windows NT 6.0; rv:21.0) Gecko/20100101 Firefox/21.0"
86.151.8.123 - - [02/Jul/2013:17:21:21 +0100] "POST /index.php?main_page=checkout%5fprocess&referer=paypal HTTP/1.0" 302 - "-" "Mozilla/5.0 (Windows NT 6.0; rv:21.0) Gecko/20100101 Firefox/21.0"

The error appears to be "checkout%5fprocess" which looks like an encoding error - can you please check with your developer on this. We're not blocking the URL at our end, but if the encoding / URL structure is incorrect, then this would lead to PayPal not be being able to access the page. 302 basically means "not found". The normal URL structure would be an underscore as in "checkout_process".

As for the IPN responses, the log shows all correct entries with 200 responses which show PayPal can access the page without any issues:

66.211.170.66 - - [02/Jul/2013:16:32:28 +0100] "POST /ipn_main_handler.php HTTP/1.0" 200 - "-" "-"
173.0.81.1 - - [02/Jul/2013:21:49:59 +0100] "POST /ipn_main_handler.php HTTP/1.0" 200 - "-" "-"

The fact that the order does not clear properly does look like a coding error on your site, with the session variables not being cleared upon order confirmation. As this is handled by checkout_process, the error above (302) may provide a clue as to what's going wrong, but you will need to look into this with your developer as we cannot provide coding support.