(1) I am busy doing a homework.
(2) I am busy to do a homework.
Victory,
You asked which of the two I would use often.
I don't think that was the question you really wanted to ask, because I don't use either of them often now that I am not in school!
Let me comment like this:
"I am busy doing homework" says what I am doing now.
"I am busy to do homework" is incorrect. It does not refer to the future. For the future, you can say, "I'll be busy doing homework" or "I'm going to do homework" or I'm going to be doing homework".
Some expressions can be followed by either a gerund ( -ing) or an infinitive (to ...). "busy" is not one of them.
_________
The -ing form is correct for both of your other examples: "Do you mind
telling me ..." and "I often see him
playing ..."
CJ