I'm studying the inference rules of natural deduction for the first-order logic. I cannot understand why universal introduction and existential elimination are valid rules. The first one says that if $\Sigma\models\theta[k/x]$ then $\Sigma\models\forall x\theta[x]$; the second one says that if $\Sigma\models\exists x\theta[x]$ then $\Sigma\models\theta[k/x]$.

Universal introduction would allow the following deduction: if $1$ is odd, then every number is odd. Existential elimination would allow the following deduction: if there exists an odd number, then $2$ is odd.

I read about some restrictions you should check before applying such rules, but they are also not so clear to me.

Some notes about the notation I used: $\Sigma$ is a set of sentences; $\theta$ is a formula such that the only free variable is $x$ and such that $x$ is not in the set of bounded variables; $\theta[k/x]$ is the formula you obtain from $\theta[x]$ by substituting every free instance of $x$ with $k$; of course, $x$ is a variable and $k$ is a constant.

Thanks.