Translation:I am proud of my son.
자랑스럽다 is more accurately translated as "to be someone's object of pride". To be proud is "자랑하다". It is same for 사랑하다 - to love, and 사랑스럽다 - to be loved by someone.
Not really. With 자랑스럽다, the thing you're proud of is to be used with 이/가 form. There are several verbs like that. I'll come back to you if/when I remember them.
Yes. This is confusing as I was not expecting the 이/가 to be the object of the verb
If that's the case, maybe a better tooltip translation for the verb would be "to make X proud". If we were talking in the third person, would the person who is proud take the objective (을/를) marker?
It's difficult to translate this to English because 자랑스럽습니다 is more of an adjective than a verb, but if you think of this way:
사랑하다 - verb - to love
사랑스럽다 - adjective - to be lovable, i.e. worthy of being loved
자랑하다 - verb - to boast
자랑스럽다 - adjective - to be "boastable" i.e. worthy of being boasted about