The use of stage terms in the closing scene of Shakespeare's Hamlet very directly reflects a line from another Shakespeare play, but it's not Romeo and Juliet, it's As You Like It:
All the world's a stage...
Existence in Hamlet is like being on a stage--everybody's acting:
- Hamlet pretends to be mad.
- Claudius pretends to be honest and just, and pretends to care about Hamlet. He toasts him in the closing scene, when he is really trying to get him to drink the poison in the cup.
- Polonius, Ophelia, Ros. and Guil. all spy on Hamlet, pretending to be his friend, or more, while actually trying to get information out of him.
- The 1 Player pretends to care about Hecuba.
- Laertes pretends to forgive Hamlet, and treats him with respect, while he is in the process of trying to kill him.
All the world, existence, is a stage in the play.
Specifically, though, the theatrical terms you mention all have literal meanings in the lines you cite. For instance, "put on," when used by Fortinbras, simply means that if Hamlet would have been "put on" the throne, he would have proved most royal.
Figuratively, I suggest the terms suggest the idea that life is a stage, and everyone acts like this or that; everyone wears masks, plays roles. This is much stronger in Hamlet than is any suggestion of drawing attention to the fact that those putting on the play are actors. That is not central to this play, as it might be in Romeo and Juliet. Why? Because there's no chorus in Hamlet. Shakespeare had matured and was past the use of a chorus by the time he wrote Hamlet. He does not go out of his way to draw attention to the actors themselves, instead of the characters, in Hamlet.