>What is that which divides a line into two lines? Is not that a point?
That's kind of cute. Its very similar to the brilliant idea of Dedekind's from which came one of the first modern developments of real numbers.
The standard technical answer to the rest of your line here is that the "ends" of a line segment are of two types: open or closed. Did you really never hear that before? An open interval has no (least,greatest) point by the usual ordering. Its hard to imagine such, and it may not even exist in physical reality, but their are ways of grappling with it successfully for the purposes of mathematics.
More fundamentally, there are some long standing philosophical questions at the heart of the issues you raise.
#1. What is the relationship between mathematics and physical reality?
#2. How do mathematicians proceed in their work these days so as to not have to bump up against #1 too often?
I have my own partial answer to #1 that the margins are too small to contain.
The answer to #2 is that they have adopted the notion of building their theories up from axioms and definitions, which more or less remove them from worrying too much about #1 and instead forces the issue to be one of the rigid "mechanistic" relationships of a formal theory. When some questions cannot be answered within such a theory there are various ways of enlarging or bifurcating the theory.