Clue vs Cue

Clue: A fact or idea that serves to reveal something or solve a problem (e.g. a crime or a puzzle).

Cue: A signal for action (like an actor entering the stage). Also, a piece of information which aids the memory in retrieving details, or indicates a desired course of action.

Really, anything can be a clue, as long as it helps to solve something. Similarly, anything can be a cue, as long as it signals meaningful information to someone. So, more than anything else, the difference is in their individual purpose, and not what they are or who might be involved.

If all that is confusing just remember this: A detective searches for clues; an actor waits for his cue.

forget doing vs forget to do

"I forgot to close the windows before getting into bed last night" means that, before you got into bed last night, you failed to recall your intention to close the windows before getting into bed, and consequently neglected to do so: you did not close the windows before getting into bed last night.

"I forgot closing the windows before getting into bed last night" means that you lost your memory of having closed the windows before you got into bed last night. The sentence may be rephrased like this: "I forgot that I had closed the windows before getting into bed last night."

The second sentence is unusual and would take a special context -- one in which you did do the indicated action (you closed the windows before getting into bed last night), then forgot about having done it, then remembered that you had done so, and then wanted to report that you had forgotten about it.

It is more common to speak of forgetting about doing something than of forgetting doing something. With "remember," on the other hand, the construction "remember doing" is perfectly fine when we wish to speak of remembering something that occurred. "Remembered to [verb]" works like "forgot to [verb]," the difference having solely to do with whether the intended action was performed.

Incidentally, I've edited your topic from "Some problems in grammar," which applies to most discussion threads on this forum, to "forget to do" vs. "forget doing," which is descriptive of the grammatical topic under discussion in this particular thread.