What is area in equation of lift?Does it remain constant during flight?
The área $A$ in lift equation $L=\frac{1}{2} \rho AV^{2}C_{L}$ is a reference area, so you can choose the one you prefer, but some election can be more apropiate than others depending on what you want to compute. For example if you are going to relate lift coefficient with drag one, the reference area should be the same, in this case a good selection will be the projected area in lift direction (aka planform area). It has the drawback that it depends on angle of attack/incidence since the true area is the same but lift vector changes. Another well selected area could be the true area (aka wetted) when working with drag, specially when friction drag is dominant considering that viscous stresses are proporcional to this area.

I give you some references when this is better explained:


  There are several different areas from which to choose when developing the reference area used in the drag equation. If we think of drag as being caused by friction between the air and the body, a logical choice would be the total surface area (As) of the body. If we think of drag as being a resistance to the flow, a more logical choice would be the frontal area (Af) of the body which is perpendicular to the flow direction. This is the area shown in blue on the figure. Finally, if we want to compare with the lift coefficient, we should use the same area used to derive the lift coefficient, the wing area, (Aw). Each of the various areas are proportional to the other areas, as designated by the "~" sign on the figure. Since the drag coefficient is determined experimentally, by measuring the drag and measuring the area and performing the necessary math to produce the coefficient, we are free to use any area which can be easily measured. If we choose the wing area, the computed coefficient has a different value than if we choose the cross-sectional area, but the drag is the same, and the coefficients are related by the ratio of the areas. In practice, drag coefficients are reported based on a wide variety of object areas.


Size effects on drag - NASA


  A comment is in order regarding the reference area Sin Eqs. (2.3) to (2.5). This
  is nothing other than just a reference area, suitably chosen for the definition of the
  force and moment coefficients. Beginning students in aerodynamics frequently want
  to think that S should be the total wetted area of the airplane. (Wetted area is the
  actual surface area of the material making up the skin of the airplane-it is the total
  surface area that is in actual contact with, i.e., wetted by, the fluid in which the body
  is immersed.) Indeed, the wetted surface area is the surface on which the pressure
  and shear stress distributions are acting; hence it is a meaningful geometric quantity
  when one is discussing aerodynamic force. However, the wetted surface area is not
  easily calculated, especially for complex body shapes. In contrast, it is much easier
  to calculate the planform area of a wing, that is, the projected area that we see when
  we look down on the wing.


Aircraft performance and design - John D. Anderson