Can we represent all algebraic structures in First-Order logic? Can we represent all algebraic structures in First-Order logic?
The subject of universal algebra is the study of algebraic structures as classes of first-order structures in languages having only function symbols as opposed to relation symbols. This subject is regarded by many as much closer to algebra than the rest of model theory, allowing one to combine methods from model theory with a stronger category-theoretic perspective. A central concern are the classes of algebras, called varieties, defined by a system of algebraic equations.

For example, Birkoff's theorem states that a class of algebraic structures is a variety of algebras if and only if it is closed under homomorphisms, subalgebras and products.

A similar corresponding theorem in first order logic is due to Kiesler and Shelah: two structures are elementary equivalent if and only if they have a common ultrapower, and one can use this to characterize which collections of algebras form an elementary class.
No, you cannot.  Fuzzy subalgebras cannot get represented in first-order logic since the background logic for a fuzzy subalgebra consists of a multivalued or infinite-valued logic.