Yes, all of these are quite closely related. I have wondered about the appropriate translation of saṅkappa, especially as the second factor of the noble eightfold path. The word intention looks to the future, which I think is appropriate for saṅkappa, related as it is to cetanā or volition, e.g. in SN12.38-12.40. However, the compound avyapāda-saṅkappa is often translated as “intention of non-ill-will”, which means the focus is no the motivation that drives the intention, not on its future direction. Do you follow me?
But that intention here should relate to motivation seems to go against the logic of the eightfold path. This logic is essentially that from right view you get right saṅkappa, and then you practice morality as a consequence. If sammāsaṅkappa refers to right motivation, however, it means that one has established mettā and renunciation before one undertakes virtue. In other words, the three factors that constitute virtue become pretty much redundant if sammāsaṅkappa refers to motivation.
I therefore think it might be better to use another translation for saṅkappa, at least in this context. The idea that it refers to the future is no doubt correct, and I would therefore suggest something like “right aim” or “right purpose”. The meaning of this is very close to “right intention”, but when you translate compounds such as avyapāda-saṅkappa the meaning is very different: “the aim of non-ill-will” or “the purpose of non-ill-will”. This seems to fit much better with how the eightfold path works. You have right view, and as a consequence you start aiming in the right direction. The first result of this aiming is moral behaviour (right speech, right action, and right livelihood), and a further consequence is the purification of the mind (right action and right mindfulness).
And I think this fits very well with what you are suggesting here. Right aim obviously includes thinking – vitakka and vacīsaṅkhāra – and it also fits as the last vestige of mental movement found in the first jhāna. Volition must be an important aspect in all these mental phenomena, because they all involve the movement of the mind (but they are also more than mere volition). They cannot exist without volition. I think this is probably why the Abhidhamma classifies so many of these terms under saṅkhāra, which in the suttas is synonymous with cetanā.
Yes, I like this phrase from AN3.101. The meaning is that the mind is no longer controlled by saṅkhāra (volition), but, as you say, on auto-pilot. Despite the residual movement of the mind, there is no sense of being in charge any more.
Sylvester off the mark? Come off it!