Locating and Copying SharePoint Customizations

I am in the process of migrating a 2010 farm from a hosted vendor to Azure (lift-n-shift). I have built the new farm and have migrated over site templates and farm solutions. However, I know there are additional customization's, perhaps located in the hive of the current farm located off-site with a Hosting company. For example, after using Sharegate to migrate the sites, the master page and layouts are not displaying and neither is the custom navigation menu. 

I don't have access to the current production servers, but I can request the current hosting company to send me whatever I need. I just don't know how to find the missing components and don't know what to copy from the current production servers. I will be eventually using a database attach to perform the FULL migration and will then use Sharegate to catch any incremental changes prior to moving to production. I have been using Sharegate to try to determine what issues I am going to have when I do the db attach. Using Sharegate, I believe, is helping me setup the web apps with the required features, etc., but I KNOW I am missing artifacts.

Any insight would be greatly appreciated. 

- Rick