There were 2 issues with the site:
1. 500 internal server error:
After debugging this issue here is what I found:
Actually, a file wp-includes/functions.php on the site was broken. My assumption is that WordPress was being upgraded to the latest version and the file was not copied completely while upgrading. After upgrade the current version of WP is 3.9.1. So, what I did, I downloaded the new version of WP 3.9.1 on my system and compared the broken functions.php with the new functions.php in my local set up and found that some piece of codes were missing, so I uploaded the new functions.php under wp-includes directory and renamed older (broken) one to 'functions-old'. The 500 error is not shown now.
I would like to delete that broken file though. I think no one would have done changes in that core file, so client will not loose any code, hopefully. It is not theme's functions.php rather WP functions.php, so I'm sure that no changes would be there and we can remove that but before removing I just wanted to make sure this thing from client that no edits has been done in WP core. In fact, a better idea would be to re upgrade the WP again, if in case any other file has been missed (however, I don't think so as everything is working fine now). If client wishes to re udgrade the system to avoid any kind of risk, he can do that from here http://duarto.com/wp-admin/update-core.php
2. 404 error placing bid
Site is using a custom template builder. Our shortcode is working fine with WP editor. But, User has used that shortcode in a section provided by page builder. Also, she has put some texts, escape characters i.e. ''\"',
tags. These all things are creating conflict with the code.
We have moved auction shortcode from that custom template to the WP editor and now it works.