Why does mutt retain deleted emails in view until you “sync your mailbox” using $

OfflineIMAP and isync are both programs that integrate well with mutt and will satisfy your other criteria.

OfflineIMAP is written in Python and isync in C; both are very quick.

Both programs are well documented and straightforward to set up; isync perhaps slightly easier.

There is one significant difference between the functionality of each that is worth bearing in mind: you can use mbsync (isync's executable) to repopulate a remote IMAP maildir¹ from a local copy, OfflineIMAP cannot do this. [Amendment: from v6.4.0, OfflineIMAP has the createfolders option to create folders on the remote repository].

The Arch Wiki has pages on both, with example configurations and use cases.²

I had some trouble with that too. The mailcap entry for text/html is the point to look at. With chromium it was indeed the needsterminal flag. For firefox the copiousoutput flag did the trick.

#text/html; chromium %s; needsterminal
text/html; firefox %s;copiousoutput 

If you don't already have a custom mailcap, you can create a file at ~/.mailcap for example and add the firefox line to it. Don't forget to specify the path in your .muttrc

set mailcap_path  = ~/.mailcap