Neighbor says: >Just now I want to deal only with straight-lines and points on them, as Euclid does in the Elements, without putting numbers on them.
Euclid didn't deal with point set topology as far as I know - that's more or less where your questions are leading. I suppose Euclid would say one could divide a line segment into two equal length parts, but he didn't look real closely at the endpoints, or much care about the so-called individual points in that sense. In today's viewpoint, you can divide a line segment (separate into two disjoint sets) into two segments with lengths of equal measure, but that doesn't mean they are topologically exactly the same when you care about such fine matters as the endpoints. Having one segment with a closed endpoint doesn't make it "longer" than the other, open ended segment so Euclid was right, if a bit course grained.
Neighbor says: >If I accept that between two points there is always another, then there is the aforementioned problem of division of a straight-line in half, which I cannot understand if I admit that between two points there's always another.
See my other answer about what's bothering you. You seem to think one "should" be able to "symmetrically" divide a line segment in two (topologically) equivalent parts or something like that. Why *should* one be able to do that? Why *should* one be able to trisect and angle with straight-edge and compass? Why *should* any number of mathematical absurdities actually exist?