Calendar invites are just attachments, you can handle them in external scripts and programs any way you like. Personally I just display them in plain text next to the email, but if you wanted to interface them with your calendar application I'm sure there are plenty of scripts doing that out there.
As for HTML email, what I take from this entire thread is that I either I've been extremely lucky so far or people in general vastly overestimate the importance of it. For one thing, most decent services will send multipart/alternative emails that are suitable for plain text display.
For instance Amazon's emails use rich HTML features, but they also have a text/plain alternative that works perfectly in a text client and carries the same informations. And when there's only HTML the vast majority of the time it's still mostly text with some light styling and it works just fine (sometimes better) in text mode.
The main problem working in big companies and using mutt as my client is that most of the time those systems have a 2nd class non-compliant IMAP servers and you have to go through oops to reliably receive and send your email. I always managed to get it to work after some tweaking though.