Considering the nature of your question, you might be interested in the following paper by Geoffrey Hellman and Stewart Shapiro:

"The Classical Continuum without Points", The Review of Symbolic Logic, vol. 6, No. 3 (2013). I will be referring to www.philsci-archive.pitt.edu/9409/1/The_Classical_Continuum_without_Points%2D_9%2D12_13th_draft.pdf

Let me quote their Abstract (from the 13'th draft) verbatim (my comments will be in square brackets):

We develop a point-free construction of the classical continuum, with an interval based on mereology and either a weak set theory or logic of plural quantification. In some respects this realizes ideas going back to Aristotle, although, unlike Aristotle, we make free use of classical "actual infinity". Also, in contrast to intuitionistic, Bishop, and smooth infinitesimal analysis, we follow classical analysis in allowing partitioning of our "gunky line" into mutually exclusive and exhaustive disjoint parts [called "intervals" which, if one wishes, may be deemed as 'points'], thereby demonstrating the independence of "indecomposability" from a non-punctiform conception [for the intervals 'cover' the gunky line]. It is surprising that such simple axioms as ours already imply the Archimedian property [for the interval structure covering the gunky line] and that they [the intervals forming that interval structure satisfying these "simple axioms"] determine an isomorphism with the Dedekind-Cantor structure of $\mathbb R$ as a complete, separable, ordered field. We also present some simple topological models of our system, establishing consistency relative to classical analysis. Finally, after describing how to nominalize our theory, we close with comparisons with earlier efforts related to our own.

I believe this paper answers your title question in the following manner (this completely my own comment):

The reason why mereology has not succeeded as an alternative to set theory is that classical analysis was, in its conception, implicitly based on the notion of 'point set'. This was realized by Dedekind, Cantor, and Frege, although the naive set theory that they used assumed that every property had an extension that was a set (engendering the Russell, Burali-Forti, and Curry paradoxes). As we know, modern set theory was developed in order to rid us of such paradoxes, but had no reason (since it was originally an analysis of the notion of 'point set') to replace the notion of 'point' (or more generally, the notion of 'element') in its axiomatization, since most "ordinary mathematicians" have not found problems with the classical conception of analysis on which modern analysis is based.

As for your second question, I believe the Hellman-Shapiro axiomatization of the gunky real line is adequate (as they show in their paper) for the development of real analysis.

Finally, a conjecture (which, hopefully, is not out of place here). Since the Hellman-Shapiro axioms derive an interval structure which is isomorphic to the Dedekind-Cantor structure of $\mathbb R$ as a complete, separable, ordered field, it seems reasonable to infer that the cardinality of the class of these intervals is $2^{\aleph_0}$. Is it possible to define such isomorphic interval structures on the gunky real line (in their notation, $G$) that are also of cardinality $\aleph_1$, $\aleph_2$,... respectively? If so, then the mereology of the Hellman-Shapiro variety will have done much to explain why there can exist such a plethora of continua, and how such can come to be. But this is, of course, mere speculation....

Considering the nature of your question, you might be interested in the following paper by Geoffrey Hellman and Stewart Shapiro:

"The Classical Continuum without Points", The Review of Symbolic Logic, vol. 6, No. 3 (2013). I will be referring to http://philsci-archive.pitt.edu/9409/

Let me quote their Abstract (from the 13'th draft) verbatim (my comments will be in square brackets):

We develop a point-free construction of the classical continuum, with an interval based on mereology and either a weak set theory or logic of plural quantification. In some respects this realizes ideas going back to Aristotle, although, unlike Aristotle, we make free use of classical "actual infinity". Also, in contrast to intuitionistic, Bishop, and smooth infinitesimal analysis, we follow classical analysis in allowing partitioning of our "gunky line" into mutually exclusive and exhaustive disjoint parts [called "intervals" which, if one wishes, may be deemed as 'points'], thereby demonstrating the independence of "indecomposability" from a non-punctiform conception [for the intervals 'cover' the gunky line]. It is surprising that such simple axioms as ours already imply the Archimedian property [for the interval structure covering the gunky line] and that they [the intervals forming that interval structure satisfying these "simple axioms"] determine an isomorphism with the Dedekind-Cantor structure of $\mathbb R$ as a complete, separable, ordered field. We also present some simple topological models of our system, establishing consistency relative to classical analysis. Finally, after describing how to nominalize our theory, we close with comparisons with earlier efforts related to our own.

I believe this paper answers your title question in the following manner (this completely my own comment):

The reason why mereology has not succeeded as an alternative to set theory is that classical analysis was, in its conception, implicitly based on the notion of 'point set'. This was realized by Dedekind, Cantor, and Frege, although the naive set theory that they used assumed that every property had an extension that was a set (engendering the Russell, Burali-Forti, and Curry paradoxes). As we know, modern set theory was developed in order to rid us of such paradoxes, but had no reason (since it was originally an analysis of the notion of 'point set') to replace the notion of 'point' (or more generally, the notion of 'element') in its axiomatization, since most "ordinary mathematicians" have not found problems with the classical conception of analysis on which modern analysis is based.

As for your second question, I believe the Hellman-Shapiro axiomatization of the gunky real line is adequate (as they show in their paper) for the development of real analysis.

Finally, a conjecture (which, hopefully, is not out of place here). Since the Hellman-Shapiro axioms derive an interval structure which is isomorphic to the Dedekind-Cantor structure of $\mathbb R$ as a complete, separable, ordered field, it seems reasonable to infer that the cardinality of the class of these intervals is $2^{\aleph_0}$. Is it possible to define such isomorphic interval structures on the gunky real line (in their notation, $G$) that are also of cardinality $\aleph_1$, $\aleph_2$,... respectively? If so, then the mereology of the Hellman-Shapiro variety will have done much to explain why there can exist such a plethora of continua, and how such can come to be. But this is, of course, mere speculation....