corfu VS company-quickhelp

You could try corfu. It has auto-completion disabled by default and works most places with M-Tab. Or you could set a longish corfu-auto-delay (0.75s say), and let auto popup only when you stop to ponder a while.

Corfu supports this. Or at least it allows you to move completions to the mini buffer which I find quiet useful.

A substantial section of the community is using corfu instead of company, but I wouldn't say company is out of date by any means. In emacs 29 eglot will be a built in, which might act as a replacement for lsp-mode depending on what functionality you need.

Wow, interesting that my response is getting down voted. It seems not enough that I give away my work for free. Nevertheless I appreciate support from the community, as other Emacs package developers. The support is actually helpful. To clarify, publishing my configuration would translate into quite a bit of work, requiring separation of private and public bits.

I am currently using evil and corfu, but I want to configure such that pressing a single escape will invoke corfu-quit when its UI is active (because pressing "C-g" is harder).

A few days ago I found an unexplained slowness when using Corfu on Emacs 29 with pgtk. I stripped my configuration to basically just Corfu, and still had a noticeable delay. Maybe it is related to your problem. https://github.com/minad/corfu/issues/307

The most straight forward solution is probably the package company-math. (that's what I use but with corfu and cape)

See here: https://github.com/minad/corfu/issues/231

https://github.com/company-mode/company-quickhelp or C-d to pop up in a separate buffer.

I'm thinking that this may be of interest: https://github.com/company-mode/company-quickhelp

https://github.com/company-mode/company-quickhelp ^ This extra company-quickhelp package looks useful, but I want to check if Eglot has it's own built-in UI functionality to implement stuff like this?

For elisp it works out of the box and displays the first line of the helpbuffer in the echo area. If you add company-box or company-quickhelp you'll get the description text next to completion candidates.