If / Whether -- Can / May: Are they interchangeable?

If and whether and can and may are not interchangeable in serious speaking and writing.
Yes, yes... I know that native speakers, even academics, have adopted of late an irresponsable usage of may/can and if/whether lowering the effectiveness of language and the clarity of communication. But that is their problem, not ours. We will stick to effective English.

IF, si, suponiendo que, a condición de, aunque, incertidumbre: Drink if you want to. If he comes I will tell him. A small cake if very tasty. His career is full of ifs.
WHETHER, si, indicando duda, posibles alternativas: Tell me whether you´ll come to my party. I ignore whether he is rich. See whether she has finished the task.

CAN, poder, saber: Can you do it? Can she speak Farsi? I can play the piano.
MAY, poder, permiso, May I come in? You may not touch that. May I ask you a question?
We have the old example of  the nine-year old Bobby who asks Johnny´s mother: "Can Johnny come out and play?" To which the mother says: "Yes, he can, but he may no." He is able to go and play but has no permission to do so.

When I hear "Can I come in?" I always answer: "I don't know. Are your legs ok?"