Can I use Fourier transforms instead of Laplace transforms (analyzing RC circuit)?
Both transforms have a large overlap in their applications. So you can use both to analyze an RC circuit. However, with the unilateral Laplace transform it's much more straightforward to take initial conditions into account, such as an initially charged capacitor. This has to do with the unilateral Laplace transform of the derivative of a function:

$$\mathcal{L}\{x'(t)\}=s\mathcal{L}\{x(t)\}-x(0^-)$$

with the initial condition $x(0^-)$ occurring explicitly.

On the other hand, if you're interested in the behavior of the circuit when exposed to periodic input signals, then it's most natural to use the Fourier transform. Note that periodic signals (extending from $-\infty$ to $+\infty)$ don't have a Laplace transform.