You are right, unfortunately MS has remove quite a lot of the useful features. If you use a lot of the missing features, it may be the best solution to stay with CE 7. Main features added in WEC 2013 are VS2013 / VS2015 support and multicore support for .NET. If you not really need this, why switching to WEC 2013?
Q: What is the recommended way to copy files to a Colibri running WEC2013 since mobile device center isn’t supported?
A: You can deploy the files in the Project Properties → Deployment → Aditional files. It has a special syntax, just copy it from an existing setup.
Q: What is the recommended way to browse the remote file system of a Colibri running WEC2013 from my PC?
A: Currently there is no out of the box solution. As this normally is only used for development, you could for example use a small FTP server on your target device. You could also use the mass storage device as described here. This feature will be available in imag V2.0 beta 1 release.
Q: What is the recommended way to remotely edit the CE registry of a Colibri running WEC2013 from my PC?
A:You should still be able to do this over the ethernet using the remote tools.
Q: What is the recommended way to use remote display to a Colibri running WEC2013 on my custom base board which doesn’t have an Ethernet port?
A: There is no out of the box solution for this.