Yes.
These items will work together as they affect the fall in different ways.
The Mushroom Vest will reduce the amount of fall damage i.e. the number of dice used to calculate damage taken.
The Boots of the cat affect the amount of damage taken when those dice for falling damage are rolled.
Generally speaking in pathfinder items effects will combine if they affect a roll or situation in different ways or provide bonuses from different sources.
So, by example, for a fall of 60 feet when using both items it would be as if you had fallen 40 feet, 4d6 would be rolled and you would take 4 damage.
You'd be better off with a ring of feather fall (Cost 2200 gp), which would reduce ALL damage for at least [as discussed in another question] 60 ft, for only 700 gp more.
Combining all three items for another example, falling 200 feet.
When falling more than 5 feet the ring will automatically activate and you will float down 60 feet leaving 140 feet to plummet.
The vest then reduces this by 20 feet, leaving 120 feet (12d6 damage)
The boots of the cat would mean this 12d6 delivers 12 damage and you land on your feet (the extra nice thing about the boots of the cat, as taking lethal from falling would normally knock you prone.)
The ring also might re-trigger (see discussion linked), reducing it further. Alternatively crafting it with a higher caster level will make it last for a longer duration.
Fluffwise:
Combining the boots of the cat and the mushroom vest I'd play it as when the player hit the floor they land on the mushroom vest which squashes and absorbs the impact, however it then springs up as the boots of the cat instantly flip them onto their feet, so it looks like they land flat and then immediately spring to their feet in a suitably dramatic pose.
No, damage reduction still applies
What they are referring to is the fact that precision damage now affects constructs, plants, and undead creatures now; in 3.5 those types were immune. In Pathfinder, only elementals and oozes are immune to precision damage.
Precision damage is of the same type as the attack to which it is applied. The only exception to this that I am aware of is attacks that deal ability damage or drain, or energy drain. In 3.5, precision damage that triggered on such attacks was dealt as Negative Energy damage. I don’t know if Pathfinder has explicitly addressed that case, but for sanity’s sake I strongly recommend all DM’s use that rule or something like it.
For the sake of citation, the rules say the following:
The rogue's attack deals extra damage anytime
That is, Sneak Attack provides a bonus to your damage roll. As a bonus to a roll, it’s just a number that gets added on to your overall damage. It does not change the attack’s damage type, it simply increases the amount of damage of that type that you do. The term “precision damage” is not a damage type, like “slashing damage,” but rather a category that certain sorts of damage bonuses go in. In this manner, it is similar to non-lethal damage.
Note: The statement that Sneak Attack may be used more frequently is perhaps misleading
That’s not really the effect that Pathfinder’s changes had: they allow Sneak Attack to be used more broadly, but there are a number of changes in Pathfinder that make it much more difficult to accomplish multiple Sneak Attacks in one round. Unfortunately, the rogue needs to do so if he is to pose much combat threat, so this is a serious problem.
Best Answer
DR doesn't work here at all, since DR doesn't apply to environmental damage. The Boots of the Cat are fine, though, and will still cap your damage to 20 and let you land on your feet.
However, you may also have some confusion about how precisely DR or falling damage works. As a refresher, falling damage is defined with this key sentence:
I'll attempt to clarify what you might be getting confused about.
You take fall damage all at once, when you finally hit the ground. You don't progressively take 1d6 fall damage per 10 feet whilst falling through the air. The increasing number of damage dice is not taking damage from the air, but an increasing measure of how hard you're going to hit the ground when you finally reach it. Someone with 1hp will survive the entire fall, so long as they have some kind of strategy for not hitting the ground too hard when they finally reach it.
If damage reduction worked with falling damage, and you fell 200 feet: Boots of the Cat would reduce that to only 20 fall damage, and Shoanti War Paint's DR 1/— would reduce that to 19 fall damage.
If you want to take no fall damage at all, maybe you'd prefer a Wand of Feather Fall, or a Ring of Feather Falling.