Do filters make the smoke cooler in your mouth?
Depends on what you mean by cooler. I enjoy 9mm charcoal filters a lot, but those are really the only filter types that appeal to me much, after smoking every kind out there over the years. A 9mm charcoal filter will take a good bit of the acidic burn out of a bitey tobacco, and make it more pleasant to smoke. It also mutes the flavor a bit, but with some tobaccos this is a blessing - I look at it not as, "This is better than that", but rather as a case where you adapt to get the best out of the style of pipe according to its material, filter, etc. For instance, sugary Danish-style aromatics that are hot and biting without a filter become very pleasant with one, and the slight flavor reduction actually works in their favor because it cuts down on the sugar overload in a lot of flavored tobaccos to make them more enjoyable. When I lived in Europe, I had a favorite Stanwell aromatic that I'd only smoke with a 9mm filter pipe, because it was much better that way AND it was a crowd pleaser that was hard to beat (smelled like Christmas ginger cookies).
The only type of filter I would advise to actively avoid are those metal stinger things, which are just a pain and offer zilch in smoking improvement. 9mms are my favorites, balsas second.