Another interesting idea to pursue might be to have modular UIs.
Instead of splitting up the current Emacs frame, I think a minimalist,
more edebug-like interface would be nice, where variable values could be
displayed using overlays instead of having a separate buffer. 

>> Here are a few comments and suggestions I found from a brief skim over
>> the code:
> I made some updates based on your suggestions and added a todo for
> Compat. 

Note that you don't need to add Compat just for `defvar-keymap', as you
could also just use the traditional method of defining a keymap instead:

--8<---------------cut here---------------start------------->8---
(defvar dape-global-map
  (let ((map (make-sparse-keymap)))
    (define-key map "d" #'dape)
    ;; ...
    map))
--8<---------------cut here---------------end--------------->8---

>         Couldn't bring myself to fix all of the checkdoc stuff, but made
> some improvments.

It is not urgent or in any way blocking inclusion to GNU ELPA, it is
just something I think one should keep in mind in the long-term to make
maintaining and contributing to the package easier.

>> Also, sorry for bringing this up, but how married are you to the name?
> I'm not overly attached to it. What are your objections? And do you have
> any suggestions? I find it quite difficult to name things like this.

It just doesn't say that much, and I don't know if it is intended, but
the usual way I would read/pronounce it (rhyming with cape) wouldn't
immediately signal any relation to DAP.  Note that you can use (elisp)
Shorthands to avoid writing out a longer name inside the file, in case a
longer name like debuger-adapter.  It seems there is no package by the
name of "dbg"?  If we are not interested in a self-descriptive and
memorable name, and would want to create a parallel to Eglot (IIRC Emacs
polyGLOT), perhaps something like based on "Emacs Debugger" (Egger?
Ebugger?  Edebugger?) might be possible as well.