Now, I know that equations are generally thought of as relationships between variables. They can refer to functions (in the case of differential equations), numbers (in the case of algebraic equations), or boolean values (in the case of logical equations). However, can one say something like:
"Let $a$ be an arbitrary differential equation and let $y$ be the set of solutions that satisfy it."
Of course, the above is just an example. Furthermore, there are many different types of equations. Is there any standard sense of algebra for equations? I know that equations can obviously be manipulated in numerical algebra, but is there a more direct sense of algebraic manipulation of actual equations rather than numbers in equations (i.e. some kind of operations upon equations and the manipulation of them).
I know this is kind of two questions in one, but I think they kind of go hand in hand. After all, if equations can be referred to by variables then one should naturally conclude how those variables can be manipulated rigorously in an operational setting.
Yes. Variable usage in the math literature world is diverse and flexible. However, most of what we wish to say can be accomplished by only referring to the objects on both sides. For example, a statement about equations of the form $f = g$ where $f$ and $g$ are polynomials, may need to refer to $f$ and $g$. Referring to "the equation $E$", while possible, may distract from a statement that's really about $f$ and $g$.
Also, when operating on equations in your sense, the equations themselves can often be encoded into objects that don't look like equations. For example, a system of linear equations can be encoded as a matrix of coefficients $A$ together with constant vector $\overrightarrow{c}$. Row reduction can be accomplished by multiplying the matrix $A$ by special row reducing matrices. The properties of those matrices are useful for results about normal forms and the like. We could restate all of those results as being about sets of equations instead of matrices, but the two ways of talking would be equivalent. Row reduction can sensibly be defined in terms of linear combinations of equations. In a sense, we already do what you are suggesting, but usually through encoding equations as other kinds of objects that have well-known algebraic operations and properties.