Use the Pythagorean theorem to find the value of x.

For a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

$$(x+12)^2 + (x+5)^2 = (x+14)^2$$

Expand each term:

$$(x^2 + 24x + 144) + (x^2 + 10x + 25) = (x^2 + 28x + 196)$$

Combine like terms on the left side:

$$2x^2 + 34x + 169 = x^2 + 28x + 196$$

Rearrange the equation to form a quadratic equation:

$$2x^2 - x^2 + 34x - 28x + 169 - 196 = 0$$ $$x^2 + 6x - 27 = 0$$

Factor the quadratic equation:

$$(x+9)(x-3) = 0$$

This gives two possible values for $x$:

$$x = -9 \quad \text{or} \quad x = 3$$

Since side lengths must be positive, we check both values.
If $x = -9$:
Height = $x+5 = -9+5 = -4$. This is not possible as a length cannot be negative.
If $x = 3$:
Base = $x+12 = 3+12 = 15$
Height = $x+5 = 3+5 = 8$
Hypotenuse = $x+14 = 3+14 = 17$
These side lengths are positive, so $x=3$ is the correct value.
Let's verify with the Pythagorean theorem: $15^2 + 8^2 = 225 + 64 = 289$. And $17^2 = 289$. So, $15^2 + 8^2 = 17^2$.

The area of a right triangle is given by the formula:

$$\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$$

Using the side lengths calculated with $x=3$:

$$\text{Area} = \frac{1}{2} \times 15 \times 8$$ $$\text{Area} = \frac{1}{2} \times 120$$ $$\text{Area} = 60$$

Angle $a$ is the angle at the base. We can use the tangent function:

$$\tan(a) = \frac{\text{opposite}}{\text{adjacent}}$$ $$\tan(a) = \frac{8}{15}$$

To find angle $a$, take the inverse tangent (arctan):

$$a = \arctan\left(\frac{8}{15}\right)$$

Using a calculator:

$$a \approx 28.07^\circ$$

The area of the triangle is $\boxed{\text{60}}$ and angle $a$ is approximately $\boxed{\text{28.07 degrees}}$.