Testing whether two elements of $\text{SL}(2, \mathbb{F}_{2^n})$ generate the entire group
The answer to your question is yes.
The subgroups of ${\rm SL}(2,q)$ were classified by Dickson in 1901 (and probably earlier). For more general questions about subgroups of ${\rm GL}(n,q)$, there is a theorem of Aschbacher that classifies them into nine types, and is used as the basis of algorithms for identifying these subgroups.
Using these results, we find that a proper subgroup of ${\rm SL}(2,q)$ with $q=2^n$ must be either reducible, imprimitive and isomorphic to $D_{2(q-1)}$, semilinear and isomorphic to $D_{2(q+1)}$, or conjugate to a subgroup of ${\rm SL}(2,r)$ where $r = 2^m$ with $m|n$.
For testing reducibility there is the efficient $\mathtt{MeatAxe}$ algorithm.
There are general tests for imprimitivity and semilinearity, but in this case it would probably be easiest to find look for an abelian subgroup of index $2$.
As YCor mentioned in the comments, you can test for definability over a subfield. Usually you start by calcuating the characteristic polynomials of lots of random group elements, and if their coefficients all lie in a subfield then it is is very likely that the whole group does. There is a method of finding the conjugating matrix to verify this, but I would have to look that up.
There are two software packages with extensive facilities for these types of calculations, GAP and Magma. I am more familiar with Magma, so I will mention some of the commands that you could use to do this. $\mathtt{IsIrreducible}$, $\mathtt{IsPrimitive}$, $\mathtt{IsOverSmallerField}$, and and $\mathtt{IsSemiLinear}$ do what you might expect.
It is possible that if $q$ is very large then you could run into some problems related to discrete log calculation, but I don't think so if you just want a yes or no answer.
The answer is yes, it is easy to check whether the ordered pair $(A, B)$ generates $\text{SL}_2(\mathbb{F}_q)$ for $q$ the power of a prime number. Indeed, there exist simple criteria according to Daryl McCullough and Marcus Wanderley, see [1, Section 11]. (Italic means that I am quoting the authors).
Claim. The pair $(A, B)$ generates $\text{SL}_2(\mathbb{F}_q)$ if and only if $$\text{Tr}(A, B) \Doteq (\text{Tr}(A), \text{Tr}(B), \text{Tr}(AB))$$ is an essential triple, i.e., doesn't satisfy any of the conditions $(1) - (5)$ of [1, Section 11].
For instance, the condition $(1)$ for a triple $(\alpha, \beta, \gamma) \in \mathbb{F}_q^3$ holds if at least two of $\alpha, \beta$ and $\gamma$ are zero. The image in $\text{PSL}_2(\mathbb{F}_q)$ of a pair $(A, B) \in \text{SL}_2(\mathbb{F}_q)^2$ such that $\text{Tr}(A, B) = (\alpha, \beta, \gamma)$ generates a dihedral group in this case. Condition $(2)$ holds if $\alpha^2 + \beta^2 + \gamma^2 - \alpha \beta \gamma - 2 = 2$, the left-hand side being the Fricke polynomial. This corresponds to the case of an affine subgroup of $\text{PSL}_2(\mathbb{F}_q)$, that is a subgroup which is conjugate to a subgroup of the image in $\text{PSL}_2(\mathbb{F}_q)$ of the subgroup of upper triangular matrices in $\text{SL}_2(\mathbb{F}_q)$. Condition $(4)$ holds if $\alpha, \beta$ and $\gamma$ lie in a proper subfield of $\mathbb{F}_q$. The remaining conditions are presented in the Addendum below.
It follows from the claim that OP's decision problem boils down to:
- deciding whether two elements of $\mathbb{F}_q$ are equal.
- deciding whether $k$ elements of $\mathbb{F}_q$ ($k \le 4$) generate a proper subfield.
- deciding whether an element of $\mathbb{F}_q$ belongs to a subfield of $\mathbb{F}_q$ generated by $k$ given elements ($k \le 4$).
The above claim follows from a theorem of Macbeath [1, Theorem 8.2] while the criteria are based on Dickson's Theorem [1, Theorem 6.1] and another theorem of Macbeath [1, Theorem 8.1].
Side note. Let $G$ be a finite group generated by $k$ elements. Let $\varphi_k(G) = \frac{\vert \text{Epi}(F_k, G) \vert}{\vert \text{Hom}(F_k, G) \vert}$, where $F_k$ denotes the free group on $k$ generators. The ratio $\varphi_k(G)$ is the probability that two uniformly chosen random elements generate $G$. By a theorem of Liebeck and Shalev, we have $\lim_n \varphi_2(G_n) = 1$ for any sequence of non-isomorphic finite simple groups, see [3, Theorem 1.1.1]. For a simple group $G_n(q)$ of Lie type of untwisted rank $n$, we have $\varphi_2(G_n(q)) = 1 - O(\frac{n^3 \log^2(q)}{q^n})$ by results of Kantor, Lubotzky, Liebeck and Shalev, see [3, Theorem 1.1.2].
Addendum. Let $Q(\alpha, \beta, \gamma) \Doteq \alpha^2 + \beta^2 + \gamma^2 - \alpha \beta \gamma - 2$ be the Fricke polynomial. This name was coined because of Fricke's trace identity $$\text{Tr}([A, B]) = Q(\text{Tr}(A), \text{Tr}(B), \text{Tr}(AB)).$$ Here are the five conditions derived by Daryl McCullough and Marcus Wanderley from Dickson's Subgroup Classification Theorem and Macbeath's Fricke polynomial criterion:
$(1)$ Dihedral case. At least two of $\alpha, \beta$ and $\gamma$ are zero.
$(2)$ Affine case. $Q(\alpha, \beta, \gamma) = 2$.
$(3.A_4)$ The elements $\alpha, \beta$ and $\gamma$ lie in $\{-1, 0, 1\}$ and $Q(\alpha, \beta, \gamma) = 0$.
$(3.S_4)$ The elements $\alpha, \beta$ and $\gamma$ lie in $\{\pm 1, \pm \sqrt{2}\}$ and $Q(\alpha, \beta, \gamma) = 1$, where $\sqrt{2}$ denotes one root of $X^2 - 2$.
$(3.A_5)$ It's complicated! The authors redirect us to [2, Main Theorem (3)].
$(4)$ Affine or $SL_2(q')$. The elements $\alpha, \beta$ and $\gamma$ lie in a proper subfield of $\mathbb{F}_q$.
$(5)$ Affine or $SL_2(q')$ or $PGL_2(q')$. The integer $q$ is odd, $\alpha^2,\beta^2,\gamma^2$ and $\alpha \beta \gamma$ lie in a proper subfield $\mathbb{F}_{q'}$ of $\mathbb{F}_q$ and at least one of $\alpha, \beta$ and $\gamma$ does not lie in $\mathbb{F}_{q'}$.
The cited article (a very nice paper!) contains more details, and of course, the corresponding proofs.
[1] D. McCullough, M. Wanderley, "Nielsen equivalence of generating pairs of $\text{SL}(2, q)$", 2013.
[2] D. McCullough, "Exceptional subgroups of $\text{SL}(2, F )$". Preprint available here, accessed 4 January 2013.
[3] I. Pak, "What do we know about the product replacement algorithm?", 2000.