It won't blow up but I wouldn't recommend stressing it until you remount the heat sink.
You will not need to remove the chip. You will remove the heat sink, clean the excess thermal paste from the chip (in place) and the cooler and then re-apply thermal paste and re-seat it.
I had a similar temp issue once when I forgot to remove the plastic backing from the base of my heatsink... you may want to check this as well when you remove yours. In my experience there can be a 6 degree or so variance between cores on a q6600 but temps shouldn't be anywhere near as high as they are for you.
Orcvader has already linked the mounting instructions for your cooler so you can follow those. When applying thermal paste just put a small pea sized blob in the center of your cpu and push down with the cooler - letting the pressure of the mount spread the paste. Never spread the paste manually or remove the cooler to check it has spread evenly as this will introduce air which will adversly affect temps.
Cleaning the old thermal paste can be done using either a specific TIM remover such as arctic cooling arcticlean or just some isopropyl alcohol. I have tried both and find that the former is easier / quicker but both work fine. Regarding new paste the one Stulid linked to is very good.
Overall remounting is not a difficult task, just don't rush it. It is possible you will have to remove the motherboard from your case if your cooler has a backplate and your case does not have an access hole.
Lots of info in drips and drabs but hope it is useful to you