Your questions have come up for students for centuries. Since we usually get to them when they're still young, fully mature debate is rarely engaged in, and since Euclid's Axioms are defined as "given" (as in "lets accept that...") there's no obligation to offer "proofs" about any of this low-level stuff. We're at the level of definition.
Just so you know you're not alone in imagining alternatives, a dimension theorist named Karl Menger proposed branching away from Euclideanism precisely around these definitions, suggesting a "geometry of lumps" we might define instead, one wherein points, lines, planes and polyhedrons aren't distinguished by "dimension numbers" (0,1,2,3).
Your discomfort with points of no size making lines would be replicated at the next levels, where lines of no breadth make "rafts" (planes) and cuts from planes of no thickness stack up to make whatever "3D printer"-like shapes. How could that be? How do we get "new dimensions" by packing and stacking what were ultimately no-dimensional points? In Menger's "geometry of lumps" maybe we don't need to think that way. 
Spooky Greek metaphysics (Hellenism some call it) is part of your cultural heritage and "infinity" is going to keep dogging you, so might as well make friends and learn it the way they teach it. But don't stop yourself from realizing that there are other axiomatic paradigms out there and Euclideanism is itself a finitude among other finitudes. 
Mathematics is about both/and thinking more than either/or thinking, though at any given time, you may need to wear blinders, just to keep those other ways of thinking from spoiling your concentration in the moment.
 I used Menger's essay as an underpinning for a later-developed finite / discrete geometry that likewise diverges from standard "dimension talk", Bucky Fuller's polymorphic system, wherein points are more lump-like (published in the late 1970s, early 1980s by Macmillan).
On Sun, Jan 26, 2014 at 10:44 PM, Neighbor <firstname.lastname@example.org> wrote:
> I mean to say "a line without breadth with points on it" and in the > Euclid's Elements this particular kind of lines (i.e. straight-lines) is > defined as one which lies evenly with points on itself and a point is that > of which there is no part, and my question is: Are there infinite points or > not on them? > If there is an infinite number of points on them and each point is > dimensionless (i.e. nothing, having neither breadth nor width) then the sum > of infinite points which are nothing will have as sum also nothing, but if > we admit that a point has only breadth (a very small amount but not 0), > then the sum of infinite points will also have breadth, and since they are > infinite the straight-line they made up is also with infinite breadth (i.e. > an infinite straight-line). Therefore, if we suppose that straight-lines > have infinite points on them, then all the straight-lines will be either > infinite or nothing (depending how we define a point). On the other hand as > you know, there are also finite straight lines, how can we admit their > existence if they are made up of infinite points? > Now, if we suppose that straight-lines have a finite number of points on > them, and a point is defined as having only breadth, the smallest possible > and not having parts, then a finite number of such points will make a > finite straight-line and an infinite number of them, a infinite > straight-line. > As you can see this is the only case which seems to be true, but if so, > the divisibility to infinity of a finite straight-line would also be > impossible, for dividing this we can arrive to the single point which have > no parts, as an atom defined by Democritus. Maybe I've been talking > nonsense, 'cause all these concepts puzzle me, and if you can show > something better, please, I should very much like to know what you think. >