(1) Is category theory the new language of mathematics, or recently the more preferred language?
Category theory has been proposed in 1940s and started taking over algebraic geometry and topology first in 1970s, and its application has only grown from there.
Whether it is the preferred language depends on which field of mathematics you are thinking about.
Generally, fields with an algebraic flavor prefer category theory. Examples include algebraic geometry, algebraic topology, category theory (duh), algebraic set theory, topological quantum field theory (new branch in physics), type theory.
Fields with an analysis/calculus flavor prefer set theory. Examples include basic calculus, differential equations, differential geometry, functional analysis, probability, set theory (duh).
Finally, fields with a geometric flavor seem to go via a third way, based on geometric intuition and "self-evident postulates". Examples include geometric topology (especially when low-dimensional), knot theory, general physics (I count it as the most important branch of mathematics), and some new fields that have "synthetic" in their names, like synthetic differential geometry.
(2) Recognizing that set theory can be articulated or founded through category theory (the text from Rosebrugh and Lawvere), is category theory now seen as the foundation of mathematics?
General mathematicians don't care much about the foundation. Currently there is a feeling that anything that has been worked on for a long time is sound. Mathematics is now seen as a massive graph of math-nuggets that connect to each other, floating in a vacuum with no special nuggets considered as "the true foundation".
However, this does not mean that all math-nuggets are equally significant, or foundation-worthy. Generally, significance is measured by these characteristics:
- How densely a nugget is connected to other nuggets;
- How "deep" the connections are;
- How fashionable it is (synthetic geometry is not fashionable now, but was extremely fashionable 2000 years ago);
- How close to physical reality it is (this makes basic calculus more significant than theory of prime numbers).
For a good discussion of what makes a nugget "significant", read Hardy's A Mathematician's Apology, starting at section 11.
As for foundation-worthiness, Maddy's What Do We Want a Foundation to Do? is a great place to learn about the details. I think that in short, some nuggets are better suited for foundation if they satisfy the following criteria:
- Can encode the things that mathematicians want to work with. This is analogous to a programming language being "Turing complete". A foundation must be "mathematician complete" or get close to it.
- Can encode elegantly. This is less objective but also very important. In the algebraic-flavored fields, category theory wins over set theory in this aspect.
- Can be checked mechanically, that is, it is good for formal verification. This is not yet a very important consideration, but is something univalent foundations and other type-theory foundations are explicitly trying to do well. For more, this Quanta essay is good for a start. For another, A computer-generated proof that nobody understands
(3) Is the choice between category theory language and set theory language maybe just depending on the field of mathematics, i.e. some fields tend to prefer set theory, others category theory?
What I said.