How to make a webpage display dynamic data in real time using Python?
To achieve your goal, you need good knowledge of javascript, the language for dynamic web pages. You should be familiar with dynamic web techniques, AJAX, DOM, JSON. So the main part is on the browser side. Practically any python web server fits. To "bridge the gap" the keyword is templates. There are quite a few for python, so you can choose, which one suites you best. Some frameworks like django bring their own templating engine.
And for your second question: when a web site doesn't offer an API, perhaps the owner of the site does not want, that his data is abused by others.
I am working on making a GUI front end for a Python program using HTML and CSS (sort of similar to how a router is configured using a web browser). The program assigns values given by the user to variables and performs calculations with those variables, outputting the results. I have a few snags to work out:

How do I design the application so that the data is constantly updated, in real-time? I.e. the user does not need to hit a "calculate" button nor refresh the page to get the results. If the user changes the value in a field, all other are fields simultaneously updated, et cetera.
How can a value for a variable be fetched from an arbitrary location on the internet, and also constantly updated if/when it is updated at the source? A few of the variables in the program are based on current market conditions (e.g. the current price of gold). I would like to make those values be automatically entered after retrieving them from certain sources that, let us assume, do not have APIs.
How do I build the application to display via HTML and CSS? Considering Python cannot be implemented like PHP, I am seeking a way to "bridge the gap" between HTML and Python, without such a "heavy" framework like Django, so that I can run Python code on the server-side for the webpage.

I have been looking into this for a quite some time now, and have found a wide range of what seem to be solutions, or that are nearly solutions but not quite. I am having trouble picking what I need specifically for my application. These are my best findings:
Werkzeug - I believe Werkzeug is the best lead I have found for putting the application together with HTML and Python. I would just like to be reassured that I am understanding what it is correctly and that is, in fact, a solution for what I am trying to do. 
WebSockets - For displaying the live data from an arbitrary website, I believe I could use this protocol. But I am not sure how I would implement this in practice. I.e. I do not understand how to target the value, and then continuously send it to my application. I believe this to be called scraping?