As for the use of the Python powered logo, again IANAL and this is not legal advice, and nor do I speak on behalf of the PSF, but right below it, the page states
In general, we want the logo to be used as widely as possible to indicate use of Python or suitability for Python. However, please ask first when using a derived version of the logo or when in doubt.
As this use case, indicating that the web backend is written in Python, seems pretty explicitly what was intended by the logo, it would seem this use is perfectly consistent with what is intended by it. However, it is certainly better safe than sorry and clarification on this point wouldn’t hurt.
If you mean setting the favicon of your website to the Python logo, I would certainly at the very least seek permission from the PSF on this point, and maybe think twice about it for your own sake. This would tend to imply not only an association with Python, but could represent your site as being an official Python/PSF website itself, given it is using their logo as yours. This could create confusion for readers over whether your website itself is affiliated, endorsed or owned by Python/the PSF, takes advantage of the PSF’s goodwill, implies endorsement of its contents and, to your own detriment, doesn’t create a unique visual identity for your own site for readers to associate with it.
Therefore, this is not legal advice, but I’d suggest rethinking this part of it and using your own unique logo—assuming I’m interpreting what you’re suggesting correctly.