Dear,
It is puzzling. I think they must be used in a suitable situation.
"I forget his name" means I usually do this.
"I forget we have homework tonight" means the agreement of the tense
"I forgot what to do"means I did the action in the past.
In my opinion;
I forget/forgot his name.
I forgot we would have homework tonight.
I forgot what to do/what I should do.
I don't think that I can persuade you like this. It is not our language.