Welcome to using HFSS!
When it comes to reproducing some published results, it is important that you have the exact setup (Apple-Apple comparison) which is usually not given in a short paper. That is why you may not be able to reproduce the exact results. But if you have enough information, you may get good agreement. Or you may reach out to the authors and ask if they can share more details about their design. I don't have access to this paper. Within the scope of the learning forum, I will provide you some answers and suggestions. Also, if you have access to customer portal, you may submit a case to work with one of our engineers in more details.
About your questions:
In HFSS we don't define a conductor as ground in a way the is defined in electrostatic. But if you are using Driven Terminal solution, when you assign port, you assign which conductor should be used as reference and so which one is the terminal. Anyways, a port should always be defined between two conductors.
The gap between the layers again is something specific to the design. The lumped port can be created in the gap between the two conductors. If one conductor is a regular trace, choose the width of the port the same as the trace. If they are pads of different width, you may choose the size of the port not too wide or two narrow as a good practice.
If the design is in nm scale, then change the Units in the Modeler (from menu) to be nm as well. Mesh operations may be needed based on the case.
Also, since you mentioned to be a new user, make sure to take advanatage of online help - HFSS comes with an extensive help document. You may also check Ansys How to videos on YouTube. There are several tutorial. In particular, I suggest you search for Exciting Microstrip with a Lumped Port in there.