Step-by-step Solution:

Step 1: Simplify the given trigonometric expression.

The expression is given by ${{2\sin P - \sin 2P} \over {2\sin P + \sin 2P}}$.

We use the double angle identity for sine, which is $\sin 2P = 2\sin P \cos P$. Substituting this into the expression, we get:

$\frac{2\sin P - 2\sin P \cos P}{2\sin P + 2\sin P \cos P}$

Since $P$ is an angle of a triangle, $0 < P < \pi$, which means $\sin P \neq 0$. We can factor out $2\sin P$ from both the numerator and the denominator:

$\frac{2\sin P(1 - \cos P)}{2\sin P(1 + \cos P)} = \frac{1 - \cos P}{1 + \cos P}$

Next, we apply the half-angle identities: $1 - \cos P = 2\sin^2(P/2)$ $1 + \cos P = 2\cos^2(P/2)$

Substituting these identities into our simplified expression:

$\frac{2\sin^2(P/2)}{2\cos^2(P/2)} = \tan^2(P/2)$

So, the problem is now reduced to finding the value of $\tan^2(P/2)$ for the given triangle.

Step 2: Calculate the value of $\tan^2(P/2)$ using the side lengths.

The side lengths of the triangle $PQR$ are given as $a=2$, $b = \frac{7}{2}$, and $c = \frac{5}{2}$.

First, we calculate the semi-perimeter, $s$:

$s = \frac{a+b+c}{2} = \frac{2 + \frac{7}{2} + \frac{5}{2}}{2} = \frac{2 + \frac{12}{2}}{2} = \frac{2+6}{2} = \frac{8}{2} = 4$

Now, we find the values of $s-a$, $s-b$, and $s-c$: $s-a = 4 - 2 = 2$ $s-b = 4 - \frac{7}{2} = \frac{8-7}{2} = \frac{1}{2}$ $s-c = 4 - \frac{5}{2} = \frac{8-5}{2} = \frac{3}{2}$

The formula for the tangent of a half-angle in a triangle is given by:

$\tan(P/2) = \sqrt{\frac{(s-b)(s-c)}{s(s-a)}}$

Squaring both sides, we get:

$\tan^2(P/2) = \frac{(s-b)(s-c)}{s(s-a)}$

Substituting the calculated values:

$\tan^2(P/2) = \frac{(\frac{1}{2})(\frac{3}{2})}{4(2)} = \frac{\frac{3}{4}}{8} = \frac{3}{32}$

Thus, the value of the original expression is $\frac{3}{32}$.

Step 3: Evaluate the given options in terms of the triangle's area, $\Delta$.

To evaluate the options, we first need to find the area of the triangle, $\Delta$. We can use Heron's formula:

$\Delta = \sqrt{s(s-a)(s-b)(s-c)}$

Substituting the values:

$\Delta = \sqrt{4 \cdot 2 \cdot \frac{1}{2} \cdot \frac{3}{2}} = \sqrt{4 \cdot 1 \cdot \frac{3}{2}} = \sqrt{6}$

This gives us $\Delta^2 = 6$.

Now, we check each option:

A: ${3 \over {4\Delta }} = {3 \over {4\sqrt{6}}}$. This is not equal to $\frac{3}{32}$.

B: ${45 \over {4\Delta }} = {45 \over {4\sqrt{6}}}$. This is not equal to $\frac{3}{32}$.

C: ${\left( {{3 \over {4\Delta }}} \right)^2} = \frac{9}{16\Delta^2} = \frac{9}{16(6)} = \frac{9}{96}$ Simplifying the fraction by dividing the numerator and denominator by 3: $\frac{9}{96} = \frac{3}{32}$ This value matches the value we found for the expression.

D: ${\left( {{45 \over {4\Delta }}} \right)^2} = \frac{45^2}{16\Delta^2} = \frac{2025}{16(6)} = \frac{2025}{96}$. This is not equal to $\frac{3}{32}$.

Step 4: Conclusion.

The value of the expression ${{2\sin P - \sin 2P} \over {2\sin P + \sin 2P}}$ is $\frac{3}{32}$. Option C, ${\left( {{3 \over {4\Delta }}} \right)^2}$, also evaluates to $\frac{3}{32}$. Therefore, option C is the correct answer.