How can I implement a web user interface for a hardware device

I am not sure if I should ask this question here but since I need something related to software I thought of giving it a try.

I have a hardware device that has an API (dll library) to control it. I am trying to find out what is the best way to have a web user interface that can use this library and control this device. Something same as what routers usually have.

My requirements are:

1. The web user interface must work on all platforms (any operating system with a browser)
2. It must use the same api. (My API is a C++ dll compiled on windows)

Note that it is flexible to add parts to the hardware such as a raspberry pi which could act as a web server, but I have to note that the api has a lot of processing so I would need a processor with high processing performance.

My questions are:

1. How do routers implement such thing.
2. Do I have to add a web server on the hardware which hosts the web ui requests
3. Is it possible to have something like both the web server and web client on one PC in order to avoid adding a new processor on the hardware.

I tried to clarify the question as much as possible, and thank you in advance.