Nothing should be wiggling inside the PC. If that is the case, something is definitely loose. check the pins, as tat is a good idea.
As for the burning, if you've not smelled it before and are smelling it now, then something is very wrong. the first thing that came to mind for me was that it is a short in the unit. Something isn't working correctly hence the burning. What could be suspect is the heatsink since you said it was wobbly, wich definitely should not be.
Sometimes, processors could catch fire. I see that you have an intel. I've heard of them catching fire on some occasions, which is why i'm not really liking them much. you say that the burning is around the motherboard? Is your processor chip on the motherboard? If so, that could be it.
Alot of troubleshooting will need to be done to make sure you get to the bottom of this. This is my opinion as ot what could be wrong. I hope i've helped some.