The functionalities of each console browser tend to different, visual restrictions are dependent on the TERM type uses, and as you said, graphics are typically unsupported, I guess a very determined author could make use of aalib.. but none that I know of do.
Protocol specific things like HTTPS and IPv6 support are possible, I know for sure that OpenBSD's lynx fork has them. A subset of Javascript and CSS would be possible, but, there are limitations.. even the rendering of HTML is best guess.
Try out the available browsers, see how they manage the site you're trying to access..
lynx
links (has an X port too..)
& w3m.
I don't know any others off-by-heart.