[Fabian]

Disclaimer: This post doesn't contain direct answers to your questions. Reasons are stated near the end.

For 1), if I understand correctly the assumption is we either checkraise/bet or checkcall/checkcall. In that case 50% equity is the cutoff between the two lines since we simply put as much money in as possible when we're a favourite to win and as little as possible when villain is a favourite to win. If neither one of us is ever folding that's all that matters.

For 2), if we use the same assumptions, the answer is 20.83% equity ASSUMING villan folds a random 20% of his hands, which is obviously not the case. Here is the math:

x is pot equity in %

EV for checkcalling both streets: (in small bets)
(7x - 3(100-x)) / 100

EV for checkraising + betting turn: (in small bets)
(0.8x*8 - 0.8(100-x)*4 + 20*6) / 100 =
(6.4x - 320 + 3.2x + 120) / 100 =
(9.6x - 200) / 100

(9.6x - 200) / 100 = 0 (checkfolding)
200/9.6 = 20.83

Note: The ev for checkcalling is always lower than for checkraising in this case.

I think the results are quite interesting here, even though they aren't directly applicable to actual play. The trouble of course is it's more likely villan will fold the 20% of hands with the lowest equity, not a random 20%.

Hopefully this will at least be a discussion starter. That's still a very low equity needed to checkraise here.

Edit: Oh, and I misread your original post and assumed he always (of the 20%) folded to the turn bet.