The first step in the process of removing the SP2010 Integration and attempting to clean out the Integration configuration data (custom columns and features etc) is:
DO NOT retract and remove the Integration solution from all Web Applications!
The second step is to raise a Customer Support ticket and request the SharePoint 2010 Integration Removal Pack
Do not attempt to rely purely on the instructions provided in the SharePoint 2013 Integration Install guide especially section 13.1.3 Removal of the SharePoint 2010 Integration Solution. That script is out of date and from the 9.1 Patch 2 release onwards the script will be removed from that document and a link to this article will be added instead.
Section 6 in the SharePoint 2013 Integration Install guide has been renamed to Upgrading from SharePoint 2010 Integration Solution and the whole section has been updated and will also reference this article.
Most of section 6 is still applicable including the instruction to NOT retract and remove the Integration solution before running the clean-up script and the restrictions around upgrading from SharePoint 2010 to 2013/2016 especially that Integration data is NOT preserved when moving to the SharePoint 2013 Integration App product.
The SharePoint 2010 Integration Removal Pack is a compressed archive containing the latest documentation on how to perform the removal and clean-up. Please be sure to read the SharePoint 2010 Integration Clean Up Instructions document first to understand the steps required as these can be different depending if the first step above had been followed or not.
The SP2010cleanup.ps1 script has been updated to remove all the Integration columns as well as better handling of the problematic Custom Field Types such as ManagementStatus, ReadOnlyString, and ReadOnlyBoolean. Please make sure you only use the latest copy of the SP2010cleanup.ps1 script provided in the SharePoint 2010 Integration Removal Pack.
The pack contains a folder named CFTCleanUp which provides extra PowerShell scripts and files required when the Integration solution had been removed and the columns cannot be removed by using the SP2010cleanup.ps1 script on its own. The SharePoint 2010 Integration Clean Up Instructions document in the pack has a section detailing how to use the CFTCleanUp files in conjunction with the SP2010cleanup.ps1 script.
This article and the SharePoint 2010 Integration Removal Pack will be update if there are still problems encountered during the clean-up operations.