Exterior derivative of a vector field The exterior derivative of a scalar function is
$d f(x,y,z) = (
  \frac{\partial f}{\partial x} dx
+ \frac{\partial f}{\partial y} dy
+ \frac{\partial f}{\partial z} dz
)$
Am I correct in assuming then that
$d\left( F_x(x,y,z) e_x + F_y(x,y,z) e_y + F_z(x,y,z) e_z \right)$
would be
$\left( 
\frac{\partial F_y}{\partial x} 
- \frac{\partial F_x}{\partial y} \right) 
dx \wedge dy +
\left( 
\frac{\partial F_x}{\partial z} - 
\frac{\partial F_z}{\partial x} \right) 
dz \wedge dx + 
\left( 
\frac{\partial F_z}{\partial y} - 
\frac{\partial F_y}{\partial z} \right) 
dy \wedge dz$

$$d(F_xdx + F_ydy + F_zdz) = dF_x \wedge dx + dF_y \wedge dy + dF_z \wedge dz$$ and use that $$dF = \frac{\partial F}{\partial x}dx + \frac{\partial F}{\partial y}dy + \frac{\partial F}{\partial z}dz$$ according to your first claim.
