- The set of rational numbers has the property described in your first question , but as you know there are holes in the set, so you cannot define continuity like that .
- Dedekind's construction of the number continuum was inspired by the continuity of straight line, so he couldn't get rid of geometric intuition completely , although he does claim that " arithmetic shall be developed out of itself", his reason is the following :
That such comparisons with non-arithmetic notions have furnished the immediate occasion for the extension of the number-concept may, in a general way, be granted (though this was certainly not the case in the introduction of complex numbers); but this surely is no sufficient ground for introducing these foreign notions into arithmetic, the science of numbers. Just as negative and fractional rational numbers are formed by a new creation, and as the laws of operating with these numbers must and can be reduced to the laws of operating with positive integers, so we must endeavor completely to define irrational numbers by means of the rational numbers alone. The question only remains how to do this.
I think this doesn't mean math based on geometry is non-rigorous , and there is no reason to get rid of geometry in analysis and stop your from using geometric intuition for inspiring foresight, things you should be aware of are geometric intuition is error-prone(learn more here, it may provide a satisfactory answer to your second question ), and algebraic method is more power than geometric methods, which caused shift of focus from geometry to arithmetic/algebra, learn more here
- Inspired by the line continuity axiom (as you mentioned it in your post), Dedekind thought a number continuum should have the property ( in his book Continuity and Irrational Numbers):
the domain R possesses also continuity; i. e., the following theorem is true: If the system R of all real numbers breaks up into two classes A1 , A2 such that every number α1 of the class A1 is less than every number α2 of the class A2 then there exists one and only one number α by which this separation is produced.
But I don't think Dedekind really achieved his goal, since he distinguish a cut produced by no rational number from an irrational number, learn more here.
Modern real analysis books just equate a cut produced by no rational number and an irrational number, so that avoid the problem Dedekind was questioned by Heinrich Weber , but which makes the real numbers a bit abstract and strange than you might think , so that Hermann Hankel made the comment (cite form here):
Every attempt to treat the irrational numbers formally and without the concept of [geometric] magnitude must lead to the most abstruse and troublesome artificialities, which, even if they can be carried through with complete rigor, as we have every right to doubt, do not have a higher scientific value.