1 He is proud of his son. why does this sentence has 'of' instead of 'from'? for me 'He is proud from his son' is more natural.
"He is proud from his son" would be unnatural and absurd, as "from" is always connected with the idea of movement from a point:
- I come from Italy/ I am from Italy (indicating origin)
You have to accept that English has been using "proud of someone/something" for hundreds of years and I would not try to teach the English people how to speak and make suggestions with prepositions you understand in the wrong way.
The same is true for your sentence no. 2. Accept that the verb construction is to convince someone of something and don't try to make English better.
In one point you are right. The use of "of" after proud is not very logical and convincing. But - and this was said already above - sometimes it is very difficult to find the reason for the use of a special preposition. And in this case I think nobody has an answer.
I might offer a wild guess. I am German and in German it is "stolz (proud) auf" and German "auf" and English "of" are very similar. Please don't understand that English today imitated a German formula. The origin of German "auf" and English "of" after proud was probably centuries or even a thousand years ago when the German variants in the North of Germany and the English variants in England were very similar.