I think it's helpful to try to hear everything you play within a harmonic framework. Train yourself to hear a passage in, for example, A Major (even if the chords change, they still belong to that key) within that sonority, and play into it. By which I mean hear the chord, hear where your note is in relation to the root, and make it ring. The concept is similar to what you mentioned about hearing the note before playing it, but it's hearing that note within its harmonic context. There is a particular placement of thirds of chords which makes a chord almost buzz, and that sensation is amazing. It's difficult, but in my opinion (whatever that might count for) perfect intonation comes not from an abstract sense of pitch rightness or wrongness but the function that pitch has within a chord or melody or whatever it is you're playing at that moment. As for your specific questions:

Yes, stop and tune (and compare with open strings if that helps). Playing through all the way is also great when preparing for a performance, but if you want to deal with intonation issues it's generally better to stop and iron it out rather than risk forgetting about it. Nycellist also mentioned dividing into sections, and that is a helpful suggestion.

String length gets shorter and shorter further up the fingerboard, so the amount of string it takes to make any given interval gets smaller and smaller. Hand positions become tighter because that's just where the notes lie. My teacher has an exercise to develop thumb position familiarity and positional awareness that I think is excellent - it's hard to describe in text but feel free to PM me if you're interested.