There is a "thing" about nose-breathing, but I have never heard that it will help you regulate HR, unless it's something like "keep your effort low enough that you can continue to breathe through your nose". But it's not the breathing, it's the effort, that affects HR.

For some with asthma or other sensitivities, nose-breathing may actually help, because it warms the air more than mouth-breathing; the idea being that warmer air does not irritate the lungs as much. But above a certain effort, it becomes very difficult for most people to maintain nose-breathing and they will switch to mouth; gets it directly in there in a hurry.

As for "filtering": I doubt nose-breathing filters anything but the largest impurities.