Eshell is interesting, but I can't use the bashisms I'm used to, and I can't copy commands back-and-forth between the prompt and a standalone script.
shell-mode lives in-between these two extremes: it runs a normal shell, but Emacs manages the buffer. I don't know about ansi-term or eshell (or alternatives mentioned by others), but shell-mode handles emoji fine, as well as progress bars and colour codes; I think it defaults to TERM=dumb, so many programs won't output colour, etc. unless you override it to something like TERM=xterm-256.
Regarding curses, I found that there was two solutions. The first is to automatically spawn curses apps in a proper terminal emulator, you just have to setup the `visual-commands` variable properly. The second alternative is to replace curses apps with Emacs apps, e.g. htop to helm-top. Personally, I ended up going the second route after a while, as I realized that there are actually very few curses apps that are important to me, and that Emacs apps are better integrated if you use Emacs for everything else.
If you rely on a lot of curses apps, a “real” terminal like emacs-libvterm may however suit you better. Renders curses apps and emojis as well as any other terminal I’ve used. It’s also much faster than ansi-term and friends at rendering.