If you had a chance to gently nudge the heatsink and it felt equally secure at all 4 corners, then the 5C difference could be a few things. Did you say that it was core0 with the highest temp? If so, then maybe it is because that is the default core for the runnning processes. Still seems strange, but IDK.
If not, then it could just be that the thermal paste wasn't applied properly. At some point when you're more comfortable, you can come back to this and re-do it.
At that point, most people here will recommend that you get an after-market fan with a back plate. You mount the back plate to the back of the mobo and the bracket to the front. Then you secure it with 4 screws.
It's more time consuming since you have to take out the mobo, but you won't have to worry about whether or not the push pins are doing their job.
btw, the different might also be due to one push pin being a little looser than the others. That combined with having a tower case where gravity is always pulling on the heatsink could be another explanation.
for now, as long as you're not going to go too wild oc'ing, it should be ok.
I don't know why the difference is less under load. You would think that it would be linear or maybe slightly non-linear with a positive slope. That's pretty interesting - at least to me. Maybe the metal cap on the chip (IHS - integrated heat sink) distributes the extra heat better - if that even makes any sense.
For fans, I like Scythe, but I think Loons are the over all favorite. Scythes are very quiet (most of them) but still move a lot of air. Check the decibel specs. For well know manufacturers, I think the numbers will be pretty accurate.
Good air flow through the case is important, but you would probably get a bigger decrease in temps by adding another fan to the heatsink. I think it's called a push-pull configuration where one fan pushes air through the HS and the other pulls it out.
You might not have room though from looking at the pics. The current HS fan looks like it's the 38mm (depth) variety.
