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$$S = t^3 - 5t^2 + 3t + 4$$ $$S(5) = (5)^3 - 5(5)^2 + 3(5) + 4$$ $$S(5) = 125 - 5(25) + 15 + 4$$ $$S(5) = 125 - 125 + 15 + 4$$ $$S(5) = 19$$

The displacement of the particle when $t=5$ is $\boxed{\text{19 m}}$.

$$V = \frac{dS}{dt} = \frac{d}{dt}(t^3 - 5t^2 + 3t + 4)$$ $$V = 3t^2 - 10t + 3$$

$$V(5) = 3(5)^2 - 10(5) + 3$$ $$V(5) = 3(25) - 50 + 3$$ $$V(5) = 75 - 50 + 3$$ $$V(5) = 28$$

The velocity of the particle when $t=5$ is $\boxed{\text{28 m/s}}$.

Set the velocity formula equal to zero.

$$3t^2 - 10t + 3 = 0$$

$$3t^2 - 9t - t + 3 = 0$$ $$3t(t - 3) - 1(t - 3) = 0$$ $$(3t - 1)(t - 3) = 0$$

Therefore, $3t - 1 = 0 \implies 3t = 1 \implies t = \frac{1}{3}$ seconds.
Or $t - 3 = 0 \implies t = 3$ seconds.
The values of $t$ when the particle is momentarily at rest are $\boxed{t = \frac{1}{3} \text{ s or } t = 3 \text{ s}}$.

$$A = \frac{dV}{dt} = \frac{d}{dt}(3t^2 - 10t + 3)$$ $$A = 6t - 10$$

$$A(2) = 6(2) - 10$$ $$A(2) = 12 - 10$$ $$A(2) = 2$$

The acceleration of the particle when $t=2$ is $\boxed{\text{2 m/s}^2}$.