Does "Doing a thing to both sides of an equation" have a name?

Is there a generic name for 'performing an operation on both sides' of an equation/inequality?

Yes. The name is algebra (maybe "doing algebra" is more appropriate and/or sounds better).

Here there are three sources:

• The word "algebra" comes from a book written in 830 by the astronomer Mohammed ibn Musa al-Khowarizmi (c. 825), titled Al-jabr w'al muqâbala. The word al-jabr meant "restoring," in this context, restoring the balance in an equation by placing on one side of an equation a term that has been removed from the other; thus if $-7$ is removed from $x^2 - 7 = 3$, the balance is restored by writing $x^2 = 7 + 3$. Al' muqâbala meant "simplification," as by combining $3x$ and $4x$ into $7x$ or by subtracting equal terms from both sides of an equation. [...] When al-Khowarizmi's book was first translated into Latin in the twelfth century, the title was rendered as Ludus algebrae et almucgrabalaeque, though other titles were also used. The name of the subject was finally shortened to algebra. (Kline's book)
• He [al-Khwarizmi] did this in his book al-jabr w al-muqabalah. “Al-jabr” (from which stems our word “algebra”) denotes the moving of a negative term of an equation to the other side so as to make it positive, and “al-muqabalah” refers to cancelling equal (positive) terms on both sides of an equation. These are, of course, basic procedures for solving polynomial equations. Al-Khwarizmi (from whose name the term “algorithm” is derived) applied them to the solution of quadratic equations. (Kleiner's book)
• The mathematical sense [of the word algebra] comes from the title of a 9th-century Arabic book ilm al-jabr wa'l-mukabala, ‘the science of restoring what is missing and equating like with like’, written by the mathematician al-Kwarizmi. (Oxford Dictionaries)

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EDIT

@BillDubuque pointed out that a correct answer for the question should give a name for the following rule: equalities are preserved by "performing an operation on both sides" (see the comments in this post).

In the Patrick Suppes terminology, the name of this rule can be Consequence of the Rule Governing Identities.

Rule Governing Identities (RGI): If $S$ is an open formula, from $S$ and $t_1=t_2$, or from $S$ and $t_2=t_1$ we may derive $T$, provided that $T$ results from $S$ by replacing one or more occurences of $t_1$ in $S$ by $t_2$. Moreover, the identity $t=t$ is derivable from the empty set of premises. (Suppes book)

Remark 1: Given an operation $f$, let $S$ be the formula $f(z)=f(z)$. Then, by the RGI, $$x=y\quad\Longrightarrow\quad f(x)=f(y).$$ In words, "equalities are preserved by performing an operation on both sides".

Remark 2: The RGI also justifies general substitutions in equalities. For example, $x+y=2$ and $x=y+3$ implies $(y+3)+y=2$ (here, $S$ is the formula $x+y=2$). To understand the word "general" see the comments in the ASKASK's answer.

Remark 3: Other names (probably, the usual ones) are Replacement, Substitution Property and Substitution by equality (see other answers and comments).

"Performing the same operation on both sides of an equality" is called the substitution property.

"Performing the same operation on both sides of an inequality" is not always possible. For instance, adding a constant is ok but multiply by one is not if the constant is negative.

For Q1, I would say that yes, roughly everything is either a substitution or an operation performed on both sides. In fact, I would argue that "an operation performed on both sides" is actually just another form of substitution.

For Q2, whenever you do something to both sides, you are just applying some function to both sides. Hence, you are using the property: $$ x=y \implies f(x)=f(y)$$

Which must hold true for all functions (by the definition of a function). For example, if you have $x^2=5$, and you squareroot both sides, you are just evaluating the function $f(x)=\sqrt{x}$ at the values $x^2$ and $5$, which must both be equal. Thus, you are simply just substituting the two values into the function and looking at the new equality you have.

EDIT just to clarify: The example of applying the squareroot function to $x^2=5$ is only valid when you note that $\sqrt{x^2}=|x|$, so the equality you are left with is $|x|=\sqrt{5}$