Yes, it is a mixed mode DLL (In fact, you can make one file of your native C++ project managed and create this C++/CLI class in that file and call the code directly from that file. You don't even need a separate DLL to accomplish this.
C++/CLI and Managed C++ both represent same thing. The only difference is that in the older version till Visual Studio 2003, it was termed as Managed C++. Later on, the syntax was changed quite a lot and it was renamed as C++/CLI. Have a look at this link for details.
Yes
CLR will be used whenever a call to the managed DLL is made.