busy+verb+ing, adjective+verb+ing?

He is busy cleaning the room.

In this sentence  "cleaning " is a gerund  and object of the verb " is ".

Likewise, Children like playing..

The gerund has several of the features of a verb.    It may have a subject of it's own. For example, There are several ways of making money.  His writing the book surprises me.

   Sometimes the passive form of the gerund phrase is used to avoid ambiguity.Example....  He told us about the hunters having been shot by them.

    The gerund may be used in the perfective.   For instance, He was suspected of having stolen the money.It may also be used in the passive.  He objected to being treated like a child.

that is called a gerund, the use of a verb as a noun.  the act to clean (verb) is being described as the action of cleaning (noun).  When the word is a thing or idea describing the process of an act rather than the act itself, it is a noun rather than a verb. It can be complicated at times.  Usually, but not always, the verb form is used to indicate the act itself (and can be an instantaneous thing), rather than the gerund which is used to indicate the continuous performance of that act, again and again and again.