### Step-by-Step Solution

#### Part 1: Finding the Average Monthly Growth Rate of Profit

Given that the profit in February was $2400 and it grew to$3456 by April, with the same average monthly growth rate for February to April, we can set up an equation to find this rate. Let $x$ represent the average monthly growth rate of profit.

The profit growth from February to April can be modeled as:
$$2400(1 + x)^2 = 3456$$

Solving for $x$:
\begin{align*}
(1 + x)^2 &= \frac{3456}{2400} \\
(1 + x)^2 &= 1.44 \\
1 + x &= \sqrt{1.44} \\
x &= \sqrt{1.44} - 1 \\
x &= 1.2 - 1 \\
x &= 0.2 \text{ or } 20\%
\end{align*}

Thus, the average monthly growth rate of profit is $20\%$. The other solution $x = -2.2$ is discarded because a negative growth rate does not make sense in this context.

Final answer for part $(1)$:
$$\boxed{20\%}$$

#### Part 2: Calculating the Expected Profit in May

Knowing the average monthly growth rate of profit is $20\%$, we can calculate the expected profit in May. The profit in April was $3456, so the profit in May, considering a$20\%$ growth rate, will be:

\begin{align*}
\text{Profit in May} &= 3456 \times (1 + 20\%) \\
&= 3456 \times 1.2 \\
&= 4147.2
\end{align*}

Therefore, the expected profit in May for the store is $4147.2$ yuan.

Final answer for part $(2)$:
$$\boxed{4147.2 \text{ yuan}}$$