1. Can add-on-generated hair be rendered with native Blender Cycles?

Yes, you can use Blender's built-in Cycles renderer to render the hair made with the add-on.
Most features and appearances will remain intact. Note:
The add-on's features like “Use Diameter Random” or “Use Radius Curve” are not supported in native Cycles;
Other structures, textures, and materials are fully compatible;
You can safely render your file on machines that don’t have the add-on installed—just avoid those special features.
📺 Demo video

2. What’s the difference between Cycles Hair Renderer and built-in Cycles renderer? Can others render hair if they don’t have the add-on?

Cycles Hair Renderer is an extended version of Cycles included in the add-on:
Retains all Cycles core rendering capabilities;
Adds advanced control for hair (e.g. diameter randomness, curve radius adjustments);
Allows more detailed hair layout control.
If someone receives a project made with the add-on but doesn’t have it installed, they can still render it using native Cycles—as long as no special add-on-only features are used. The result won't be much different.
📖 Documentation: Cycles Hair Renderer – Features & Overview