Does lambda calculus have character strings, structures, exceptions, symbols, mutable variables, quoting code as data, macros and three different kinds of object equality?
Some of the algorithms for reducing lambda expressions are symbolic, like, say, alpha reduction. But they are not done in lambda calculus itself, as far as I can see; they are done on lambda calculus (by the mathematician, or a machine for manipulating lambda calculus terms).
Analogy: just like ordinary algebra doesn't have symbols; but of course mathematicians work with symbols and so do computer algebra systems.
Since lambda calculus is Turing complete, it can of course represent lambda calculus manipulation (via Goedel numbering of lambda terms and then working with their arithmetic encoding as Church numerals, or whatever).