Looking for a good CPU cooler

Agree with everything except for the last part about removing the motherboard for an aftermarket CPU cooler, you can often just install it through the back of the case (by removing the right panel) unless the case doesn't have an opening behind the motherboard, CPU socket.

It's not hard, but becareful. Don't put pressure on the CPU when you put the CPU cooler back on top, like don't shake it left/right, it can damage your CPU pins. Keep it straight, very light pressure, and it won't be an issue.

Thermal paste amount doesn't have to be more than a rice grain on the middle of the CPU. Clean it with alcohol first.

Your temperatures are high, but not dangerous.
The throttle point is around 100c. The cpu will slow down or shut off if temperatures get that high.

45c at idle likely indicates a problem.
You should see 10-15c. over ambient if your cooler is functioning and mounted well.
Possibly, it has come loose.
Check that all 4 pushpins are through the motherboard and locked. The cooler should not wiggle if you nudge it.
See my stock pushpin mount instructions at the end.

I would invest some $30 for a cryorig H7 cooler. If nothing else, the 120mm fan will be quieter.
It is more compact and easier to mount than the hyper212 coolers.

Check my rant on liquid coolers at the end, a aio is entirely inappropriate for your needs.

----------------how to mount the stock Intel cooler--------------

The stock Intel cooler can be tricky to install.
A poor installation will result in higher cpu temperatures.
If properly mounted, you should expect temperatures at idle to be 10-15c. over ambient.

To mount the Intel stock cooler properly, place the motherboard on top of the foam or cardboard backing that was packed with the motherboard.
The stock cooler will come with paste pre applied, it looks like three grey strips.
The 4 push pins should come in the proper position for installation, that is with the pins rotated in the opposite direction of the arrow,(clockwise)
and pulled up as far as they can go.
Take the time to play with the pushpin mechanism until you know how they work.

Orient the 4 pins so that they are exactly over the motherboard holes.
If one is out of place, you will damage the pins which are delicate.
Push down on a DIAGONAL pair of pins at the same time. Then the other pair.

When you push down on the top black pins, it expands the white plastic pins to fix the cooler in place.

If you do them one at a time, you will not get the cooler on straight.
Lastly, look at the back of the motherboard to verify that all 4 pins are equally through the motherboard, and that the cooler is on firmly.
This last step must be done, which is why the motherboard should be out of the case to do the job. Or you need a case with a opening that lets you see the pins.
It is possible to mount the cooler with the motherboard mounted in the case, but you can then never be certain that the push pins are inserted properly
unless you can verify that the pins are through the motherboard and locked.

If you should need to remove the cooler, turn the pins counter clockwise to unlock them.
You will need to clean off the old paste and reapply new if you ever take the cooler off.
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My canned rant on liquid cooling:
------------------------start of rant-------------------
You buy a liquid cooler to be able to extract an extra multiplier or two out of your OC.
How much do you really need?
I do not much like all in one liquid coolers when a good air cooler like a Noctua or phanteks can do the job just as well.
A liquid cooler will be expensive, noisy, less reliable, and will not cool any better
in a well ventilated case.
Liquid cooling is really air cooling, it just puts the heat exchange in a different place.
The orientation of the radiator will cause a problem.
If you orient it to take in cool air from the outside, you will cool the cpu better, but the hot air then circulates inside the case heating up the graphics card and motherboard.
If you orient it to exhaust(which I think is better) , then your cpu cooling will be less effective because it uses pre heated case air.
And... I have read too many tales of woe when a liquid cooler leaks.
google "H100 leak"
-----------------------end of rant--------------------------

Your pc will be quieter, more reliable, and will be cooled equally well with a decent air cooler.