Quadratic Extension of Finite field

Let $q = p^n$.

The roots of the polynomial $x^q - x$ are precisely the elements of $\mathbb{F}_q$. If $\alpha \in \mathbb{F}_{q^2}$, then $\alpha^{q^2} - \alpha = 0$, and

$$ \left(\alpha^{q+1}\right)^q = \alpha^{q^2 + q} = \alpha^{q^2} \cdot \alpha^q = \alpha \cdot \alpha^q = \alpha^{q+1} $$

Therefore $\alpha^{q+1} \in \mathbb{F}_q$.

In general, given a finite extension $L/K$ of fields, there is the concept of the norm (over $K$) of an element of $L$, which is the product of all of its conjugates, and is an element of $K$. If $K = \mathbb{F}_q$, the conjugates of any element are obtained by repeatedly raising it to the $q$-th power.

In this case, we have a quadratic extension, so

$$N(\alpha) = \alpha^q \cdot \alpha = \alpha^{q+1}$$

which gives another proof that $\alpha^{q+1} \in \mathbb{F}_q$.

To see that every element of $\mathbb{F}_q$ is a norm -- that is, a $(q+1)$-th power of an element in $\mathbb{F}_{q^2}$ -- we can appeal to polynomial equations again. Every $\alpha \in \mathbb{F}_{q^2}$ is a root of some equation

$$ f_\beta(x) = x^{q+1} - \beta = 0 $$

and each such equation has at most $q+1$ roots. $f_0(x)$ has a single root $0$ with multiplicity $q+1$. The remaining $q^2 - 1 = (q-1)(q+1)$ elements of $\mathbb{F}_{q^2}$ are distributed among the polynomials corresponding to remaining $q-1$ values for $\beta$, with each $f_\beta$ having at most $q+1$ of them. A counting argument thus shows that every $f_\beta(x)$ with $\beta \neq 0$ must have exactly $q+1$ distinct roots in $\mathbb{F}_{q^2}$.

We could (as the other answers have shown) argue by using the fact the unit group of $\mathbb{F}_{q^2}$ is cyclic.

Having the idea of the norm, it's easy to generalize to arbitrary finite extensions of $\mathbb{F}_q$: the norm of an element of $\mathbb{F}_{q^n}$ is

$$ N(\alpha) = \alpha \cdot \alpha^q \cdot \ldots \cdot \alpha^{q^{n-1}} = \alpha^{(q^n-1) / (q-1)}$$

Of course, this again could have been discovered by using the fact the unit group is cyclic.

(For extensions $L/K$ of arbitrary fields, it's not always true that every element of $K$ is a norm)

Work in progress

The litmus test for deciding whether an element $\beta$ in an extension field $L = \mathbb F_{q^m}$ belongs to the base field $K = \mathbb F_q$ is to check whether $\beta^q = \beta$ or not. If $\beta^q = \beta$, then $\beta \in K$; else $\beta \notin K$. In your case, $m = 2$. Given any $\alpha \in L = \mathbb{F}_{q^2}$, write $\alpha^{q+1} = \beta$, and note that $$\beta^q = \left(\alpha^{q+1}\right)^q = \alpha^{q^2+q} = \alpha^{q^2}\cdot\alpha^q = \alpha\cdot\alpha^q = \beta$$ where we have used the litmus test $\alpha^{q^2}\stackrel{?}{=}\alpha$ for membership in $\mathbb F_{q^2}$ to aver that $\alpha^{q^2} = \alpha$. Applying the litmus test for membership in $\mathbb F_q$ to $\beta = \alpha^{q+1}$, we have that $\alpha^{q+1} \in \mathbb F_q$ for all $\alpha \in \mathbb F_{q^2}$. In other words, the function $x \mapsto x^{q+1}$ maps $L=\mathbb F_{q^2}$ to $K = \mathbb F_q.$

Turning to the matter of the orders of the elements, let $\alpha \in L$ denote a generator of $L^*$ so that ${\mathsf{ord}}(\alpha) = q^2-1$. Then $${\mathsf{ord}}(\alpha^{q+1}) = \frac{{\mathsf{ord}}(\alpha)}{\gcd({\mathsf{ord}}(\alpha),q+1)} = \frac{q^2-1}{\gcd(q^2-1,q+1)} = q-1$$ and so whenever $\alpha$ is a generator of $L^*$, it is mapped by the transformation $x \mapsto x^{q+1}$ onto $\beta=\alpha^{q+1}$ which is a generator of $K^*$: generators of $L^*$ are mapped into generators of $K^*$ Now, since $\beta$ generates $K^*$, every generator $\gamma$ of $K^*$ can be expressed as a power of $\beta$, that is, for each generator $\gamma$ of $K^*$, there exists an integer $k$, relatively prime to $q-1$, such that $\gamma = \beta^k$. Then, $$\left(\alpha^k\right)^{q+1} = \alpha^{k(q+1)} = \left(\alpha^{q+1}\right)^k = \beta^k = \gamma.$$ Is $\alpha^k$ a generator of $K^*$? Well, $${\tt{ord}}(\alpha^k) = \frac{{\tt{ord}}(\alpha)}{\gcd({\tt{ord}}(\alpha),k)} = \frac{q^2-1}{\gcd(q^2-1,k)}$$

Furthermore, we can use the fact that $\tt{ord}(\alpha) = q^2-1$ to deduce that for any integer $i$, $$\left(\alpha^{k+i(q-1)}\right)^{q+1} = \alpha^{k(q+1)+i(q^2-1)} = \alpha^{k(q+1)}\cdot\alpha^{i(q^2-1)} = \alpha^{k(q+1)} = \beta^k = \gamma.$$ Thus, the $q+1$ elements $\alpha^k, \alpha^{k+(q-1)}, \alpha^{k+2(q-1)}, \ldots, \alpha^{k+q(q-1)}$ are the $q+1$ roots of $x^{q+1} - \gamma \in \mathbb F_q[x]$. At least one of these elements must be a generator of $L^*$. Work in progress

HINT:

The multiplicative group of a finite field is always cyclic. Thus $L^\times$ is cyclic of order $p^{2n}-1=(p^{n}+1)(p^n-1)$ and $K^\times$ is reobtained back as the only cyclic subgroup of order $p^n-1$.

By writing $L^\times=\langle\gamma\rangle$ and writing every element as a power of $\gamma$ all the answers to your questions should be straightforward.