@arealsnek
Check out Fractals line of cases, avoiding anything with a "basement/psu shroud" if possible, not a good idea in terms of thermals.
Within your budget a Fractal Define R5 would probably work well for you, and it has a ton of sound deadening material, not to mention the dust filters..
It has dust filters in the front and all along the bottom.
For Reference
My Current Build is
Fractal Define R5
6700k @ 4.3Ghz w/Noctua NH-D15
Strix GTX1080 @2Ghz Powr limit @120% w/2x Noctua 140mm Fans as Cooling.
I'm Running 2x140mm fans in the front of the case, 1x140mm as a side intake, and 1x140mm fan as an exhaust out the back.
The top vents are covered up, and thermal results after a 20 minute AIda64+Furmark are as follows
GPU Peaked at 61C
CPU Peaked at 65C
Finally all fan speeds were in 600-1000rpm range, and hardly any noticeable noise.