Adjective+Preposition+Gerund?

I think with this case in particular, the rule doesn't matter so much. Both "busy with doing my homework" and busy doing my homework" are acceptable.

However, in other cases, you'd have to obey the rule.
E.g. He is afraid of losing the bet - it wouldn't make sense to say "He is afraid losing the match"
E.g. They are glad about having a baby - likewise, this sentence would lose its meaning if you omitted the preposition.

Having said that, I can think of an example where losing the preposition would be fine:
"The student is not happy about seeing the teacher" is the same as "the student is not happy seeing the teacher."

So, the need for the rule varies from case to case. Just be careful and make sure you use it when it is needed (:

I never heard of any such rule. And it's very rare to hear anybody talk that way. It's not wrong, just rare.

Doing is a gerund?? Nah.
A gerund is a verb ending in -ing, which can also be used in sentences as a noun.
Eg: Swimming, Jogging, walking, etc, which can be used like:
Walking is a good exercise.
Dieting is not a popular activity.

I am busy with = okay.
I am pre-occupied with, I am concerned about.

I have not heard about your rule.