How to properly unit test a web app?
Most of this is personal opinion and will vary from developer to developer.
There are a ton of python libraries for unit testing - that's a decision best left to you as the developer of the project to find one that fits best with your tool set / build process.
This isn't exactly 'unit testing' per se, I'd consider it more like integration testing. That's not to say this isn't valuable, it's just a different task and will often use different tools. For something like this, testing will pay off in the long run because you'll have piece of mind that your bug fixes and feature additions aren't impacting your end to end code. If you're already doing it, I would continue. These sorts of tests are highly valuable when refactoring down the road to ensure consistent functionality.
I would not waste time testing 3rd party APIs. It's their job to make sure their product behaves reliably. You'll be there all day if you start testing 3rd party features. A big reason to use 3rd party APIs is so you don't have to test them. If you ever discover that your app is breaking because of a 3rd party API it's probably time to pick a different API. If your project scales to a size where you're losing thousands of dollars every time that API fails you have a whole new ball of issues to deal with (and hopefully the resources to address them) at that time.
In general, I don't test static content or html. There are tools out there (web scraping tools) that will let you troll your own website for consistent functionality. I would personally leave this as a last priority for the final stages of refinement if you have time. The look and feel of most websites change so often that writing tests isn't worth it. Look and feel is also really easy to test manually because it's so visual.