So long as you treat k and E as static numeric values, then Hooke's law, etc. are only applicable over a narrow range of conditions, most notably temperature and whether or not the material has previously exceeded its elastic limit.

So long as the stress/strain curve remains linear, you can generally safely assume that the resultant k value remains constant so long as E remains constant.

You must also remain cognizant that another form of linearization is assuming the cross section remains the same, which it doesn't when you noticeably deform it.

With regards to divergence errors within FEA through further discretization:

If you were to calculate FEA by hand, you would find that you can transmit forces through your sharp corner. And you could then take your loads, divide by area, and then come up with a guess on the stress.

Then if you were to double that mesh density, you would find that your forces aren't converging to 0 as fast as your areas are, thus your calculated stresses go up.

Continue doubling the density, but otherwise keeping the same topology creates this divergence error.

Ideally, you would end up with a "magic mesh" where increased discretization results in a linear relationship between "F" and "A" between increases in discretization, and stresses converge to an answer.

Taking your cube for example, you know that F at end one will transmit to the other end. Thus, splitting it into smaller chunks will result in each chunk being 1/nth of the original, and summing them all back up comes to F.

But if your cube instead has a hole in the middle, if you end up with vertices at 12:00, 3:00, 6:00, and 9:00, you're going to have divergence errors, because mathematically, you can't turn a vertical load into a horizontal one.