Constantine: the point of checking the behavior of the Japanese IME was to determine whether there is a problem with all IMEs or just some.  Since Ctrl + the 2nd key from the left on the bottom row (marked 'X') invokes 'Delete' in both American and Japanese entry modes, the behavior you want should be possible.

I do not know where the bug is, and do not care, other than knowing that it is not in IDLE (or Python).  I do care that anyone in the world, like you, with sufficient knowledge to use Python, be able to also use IDLE if they chose.   Even if switching keyboards with alt-shift is a fast, habituated, motion, it is clearly a nuisance in this context  So, even though IDLE is not Firefox, I am willing to consider somehow implementing what Serhiy described and supporting either adding workaround keyset(s) to IDLE or make it easier for users to create their own custom keysets that work.

When I mentioned the menus, I assume that the alt menu shortcuts work, so alt-e, alt-c, for instance, would copy a selection.  When we tried this with the Japanese keyboard, the alt shortcuts work for the top menu (F_ile, E_dit, S_hell, etc) but not for the drop-down submenus.  After alt-e displays the edit menu, alt-c, etc, have no apparent effect.  Another bug, and one which I doubt we can work around without language-specific menus.  (This is a separate issue.)

Step 1 for this issue: what is the scope of the problem.  I presume it is not unique to Ukranian.  Does it affect all Cyrillic IMEs?  If so, do different cyrillic keyboards have the the alphabetic chars in different places, so that a separate workaround is needed for every IME, even for letter chars?