Are all the functions whose fourier weight is concentrated on the small sized sets computed by AC0 circuits? Are all the functions whose fourier weight is concentrated on the small sized sets(or terms with low degree) computed by $\mathsf{AC}^0$ circuits ?
No. Consider the following function on $\{0,1\}^n$:
$$ f(x) = x_0 \land \cdots \land x_{n-\sqrt{n}-1} \land (x_{n-\sqrt{n}} \oplus \cdots \oplus x_{n-1}). $$
Clearly this function is hard for AC0. On the other hand, the function is almost constant, so almost all of its Fourier spectrum is on the first level.

If you want a balanced counterexample, consider
$$ g(x) = x_0 \oplus \left[x_1 \land \cdots \land x_{n-\sqrt{n}-1} \land (x_{n-\sqrt{n}} \oplus 
\cdots \oplus x_{n-1})\right]. $$
This function is almost always equal to $x_0$, so almost all of its Fourier spectrum is on the first two levels.