Validity of Friis equation

Friis equation is valid only if the receiving antenna's effective area Ae is fully illuminated. Is my statement correct?

No, I don't think so.

Effective is the word. In a radio link budget for which the Friis' equation is used, if your antenna is poorly illuminated or designed, its EFFECTIVE area or gain degrades. But this is antenna problem, not Friis' formula problem.

The gain of an antenna is usually directional.
A simple gain value states the max gain of the main beam.
If you aren't using the main beam you need the gain value for the used direction.

What care should I take to correctly apply Friis equation?
Arbitrary choice of gains Gt and Gr of antennas allows Pr/Pt to take any positive real value.
Since that cannot happen, there must be some limitations on Gt and Gr. Am I right?

Gt and Gr depends on your antenna design.
For example dipole antenna: gain ~ 1.64 (2.15 dBi)
Or a patch antenna: gain ~6.3 (8dBi).
But bigger antennas have much more gain for example for satellite communication

The friis formula is valid when antenna A provides uniform illumination of antenna B. That means the beam width of antenna A should be large w.r.t. the size of antenna B. Same condition should apply for antenna B when radiating to antenna A.

Many references state that the distance should be more then the sum of the far field distances for each antenne. Far field distance (rff) is taken as rff = 2b^2/lambda.
Where b = largest size of antenna. In real world this formula is very conservative and frequently the factor 2 can be omitted.

Any reflection (within or outside the first Fresnel zone) will change the received power. A reflection that is 10 dB below the LOS beam can reduce the received power with 3.3 dB (depending on phase relation and antenna gains in the direction of the reflection).

Antennas with high gain, have size well above lambda and this gives an increase of rvv. Also antennas with large gain have small half power beam width. Just an indication: Gi = 45k/(hpbwv*hpbwh). Gi is not in dB, hpbw = half power beam width, h = horizontal plane v = vertical plane. So antenna with hpbwh = hpbwv = 10 degrees, will have Gi of around 450 (26.5 dBi).