@AntonPetrunin I see it now ,good question, I think what you mentioned may add additional info about uncount-point set for a typical continuous function that should be considered from the point of question 2.

@AntonPetrunin sorry I can't understand of what you call "typical".

@GerryMyerson personally I don't know what's wrong with the question ,I wish there be someone suggesting an edit in order to make the question in-topic.

@JoshuaZ I asked there as I mentioned but they had put on hold without any appropriate answer or any obvious reason like here .I don't know what is the problem with my question, nobody even say anything about it or edit it!

@GerryMyerson at least would you please say where should I ask my question sir?

@GerryMyerson sorry sir, I didn't get your point ,would you please explain it more?

You argued about non-intersecting situations and it seems majority of them are related to convex bodies in their non-smooth points, it is still interesting to figure out the competition between amount and treat of intersecting and non-intersecting points in more complex shapes than just convex and over different angles of rotation, for example finding shapes that your set $R(\alpha)$ becomes dense and uncountable.