Should my chimney sit on my roof?
The real question is not "Is it OK?"  The real question is "Should this be fixed?" Or perhaps "Will this fall down?"  Or maybe even "Should this have been done differently?"  Any answer you get will answer one of those questions.  "Is it OK?" is really asking "Is this something I can live with?" which is a question you have to ask yourself.

Will this fall over?  Probably not.  But I don't think anyone will give you a guarantee on that.

Should this be fixed? It depends on how the load is sitting on the roof.  Is it being supported by a load-bearing wall or is this in the middle of the roof and the roof itself is holding the load?  I'm guessing it's on an exterior wall.  So it's probably fairly stable.  I imagine that it should be replaced at some point.  That's really the only way to attack this problem: full replacement.

Should this have been done differently? Yes.  You don't fix a chimney by sealing it inside another chimney.  That would be like replacing your siding by building another wall.  It doesn't solve the root issue.

In the end, it sounds like you have a bad chimney and you will probably want to replace it at some point in the future.  Really, you should have someone come out and take a look.  It's definitely not a DIY project.

However, since you're renting, does it really matter?