Quick questions on Discrete random variables: probability distribution, expected value, variance and standard deviation

The expected value (or mean) of $X$ is the long-run average value if we repeated the experiment many times. It is the weighted sum

In the exam a discrete distribution is usually laid out as a two-row table: the values $x_i$ on top and the probabilities $p_i$ underneath. Reading it correctly is the first marked step. Check the probabilities sum to $1$ (solve for any unknown if a constant is involved), then build the calculation columns you need: $x_i p_i$ for the mean and $x_i^2 p_i$ for $E(X^2)$. Laying the work out in columns keeps the arithmetic tidy and is exactly what markers look for.

$\text{Var}(a X + b) = a^2 \text{Var}(X)$, not $a \text{Var}(X)$, and the $+ b$ has no effect on variance.

If you get a negative variance, you have a calculation error. Variance is always non-negative.