Who takes care of the stakeholders

First my standard disclaimer.  This is my interpretation and it may (and most likely will) be different than others that participate in this thread.

Product Owner is accountable for ensuring that the stakeholders are informed and included. However, I'm going to quote this line from the Scrum Guide where it talks about the Product Owner's responsibilities.

The Product Owner may do the above work, or have the Development Team do it. However, the Product Owner remains accountable.

Since part of that responsibility includes this statement

Ensuring that the Product Backlog is visible, transparent, and clear to all, and shows what the Scrum Team will work on next;

I believe that the Product Owner is responsible but that it can be done by anyone.Since the Scrum Backlog is made of Product Backlog Items, I extend that responsibility to the Scrum Backlog. I, and all of the POs I work with, encourage the development team to interact with the stakeholders continuously as it helps with the inspect/adapt philosophy.  We do mini "demos" as the sprint goes on. It is not unusual to see a stakeholder sitting at a developers desk discussing the current implementation and determining if it needs to be modified and how. In fact I, as Scrum Master, have had stakeholders sit with me and look over what has been done. (My past life is in Quality Assurance so I have some background in doing this.) 

As for inviting them to the Reviews, again why can't anyone do so? Self-organizing, self-managed teams should be equals. 

The Scrum Guide states roles that are accountable for actions/activities but it never says that only those roles can perform those actions/activities.  Just as the Scrum Master is accountable for ensuring the Daily Scrum occurs and is held to 15 minute time-box, the Scrum Master is not a participant. I usually watch from across the room and then "monitor" the work being done. If I see that there could be an issue, I'll bring it up with the team, let them know I'm going to observe and possibly offer suggestions. But there is nothing that prevents the Product Owner or even the Development Team themselves from taking part in that same activity.

Who invites them to the Sprint Review?

The Scrum Guide says: “Attendees include the Scrum Team and key stakeholders invited by the Product Owner”