Solution:
(1) Since the function $f(x)=a^2x^2$ ($a>0$), translating the graph of the function $y=f(x)$ one unit to the right yields the graph of the function $y=\varphi(x)$,
Therefore, the expression for $y=\varphi(x)$ is: $y=\varphi(x)=a^2(x-1)^2$. Given the non-negativity of a perfect square, its range is: $[0, +\infty)$.

(2) Solution 1: The solution set of the inequality $(x-1)^2>f(x)$ contains exactly three integers $\Leftrightarrow (1-a^2)x^2-2x+1>0$ has exactly three integer solutions,
thus $1-a^20$ and $h(1)=-a^20$),
one root of the function $h(x)=(1-a^2)x^2-2x+1$ is in the interval $(0, 1)$, and the other root must be in the interval $[-3, -2)$.
Therefore, $$\begin{cases} h(-2)>0 \\ h(-3)\leq 0\end{cases}$$ yields $$\boxed{\frac {4}{3}\leq a\leq \frac {3}{2}}$$.

Solution 2: $(1-a^2)x^2-2x+1>0$ has exactly three integer solutions, thus $1-a^21$.
$(1-a^2)x^2-2x+1=[(1-a)x-1][(1+a)x-1]>0$
Thus, $$\frac {1}{1-a}<x< \frac {1}{1+a}$$, and since $$0< \frac {1}{1+a}<1$$,
it follows that $$-3\leq \frac {1}{1-a}<-2$$, which yields $$\boxed{\frac {4}{3}\leq a\leq \frac {3}{2}}$$.