Let's use できる as an example.

できる (I) can do (it)/possible (action/event). できるか? Can do?/Possible? できますか?
Can do?/Possible? (polite)

Believe it or not, できますか? still sounds too direct to me.

Now here's my question, is this dekimasuka?:

(もし)できますでしょうか? Possibly can do?/Perhaps possible? and (もし)できませんでしょうか? Can't
(you) possibly do?/Isn't it perhaps possible?

If the above statements are grammatically sound, how about natively? When a
native person hears it, what is their average response to such a
statement/question?

Another related question: according to some teachers of Japanese, the か at the
end of a question could be considered harsh, too direct or unnecessary. For
example, what is the difference (if any) between [たべる?] (with question mark)
and [たべるか?] (with question mark and か)?

Optional question: Where does だろうか? and でありますか?/でありませんか? fit into all of this?
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question
grammar usage politeness
Adding でしょうか to ます to further emphasize doubt and politeness?
A plain form + か does sound a little more rough or direct, I think. Normally
it would be said without the か or with の instead as it's a little softer.
できますか, however, is not direct at all and is a totally acceptable way to ask
something politely.

In polite _keigo_ type situations, you may in fact here できますでしょうか or
いただけますでしょうか. Technically this is incorrect. You _might_ call it a form of 二重敬語
(maybe, don't quote me on that). However despite it being "wrong," you still
might hear it from time to time. Kind of like ら抜き言葉 (食べれる) or omitting the い
in ている (～してます). So yeah, it's possible to use it this way, but be aware that
doing so is breaking with convention.

The standard way to say it in these situations is, of course, plain form +
でしょう, so できるでしょうか and the like.
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answer