Yes, fire protection stacks. To test this, I put on a diamond helmet with Fire Protection I and stood in fire. Then, I put on an entire suit of diamond armor with Fire Protection I and went to stand in the fire again. I took no damage, though my armor took extensive damage.
This is the video I made in the process (I'm not sure I've properly annotated it, but hopefully you can tell what I'm doing. :P):
Protection will stack on different armor pieces, and provides comprehensive protection against all forms of damage except The Void and server kill commands. It is true that falling, fire, lava, poison, and even mob damage will be reduced, but false that the armor durability will always reduce more than normal. When falling off a building with Protection IV boots, the boots will not suffer any damage, producing the same results as feather falling III boots. The Armor will take more damage from Mobs and Players however.
It is possible to get feather falling IV AND protection IV boots, reducing fall damage immensely. All other pieces of armor with protection will add additional damage reduction in the same way the boots would. Protection appears to be one step down from specific defenses (ex: Protection III = Feather Falling II when taking fall damage, or Protection III = Fire Protection II when in lava).
TL;DR- Protection enchantments are slightly less effective than specific ones, but protect against everything at once. The best possible armor would be:
Helmet: Respiration III, Aqua Affinity I, Protection IV, Thorns III
Chest and legs: Protection IV, Thorns III
Boots: Protection IV, Feather Falling IV, Thorns III
Note that Only highest Level of Thorns take effect if present on multiple peices of armor. Source
Best Answer
Durability for armor is the amount of damage they can absorb when worn. From chand's answer:
So, better armor certainly last longer. Do they provide better protection though? It depends.
The protection factor (% of damage absorbed) is:
So a brand new full-body leather suit will give you 80% protection, just like a brand new diamond suit. However, material does make a difference in durability too — in both good and bad ways.
Example:
A half-worn leather suit with brand new iron boots (the cheapest iron armor) protects much more than a half-worn leather suit with brand new leather boots (62% vs 49%). However, a brand new leather suit with half worn iron boots protects much less than a brand new leather suit with half worn leather boots (57% vs 70%).
This is because the total armor wear is not the average of each item's wear, so the better material an item is, the more heavily it weighs in determining the overall quality. As shown, this makes things better and worse.
So, if you need a quick boost to your survivability, temporarily swap in iron armor; be aware that once it starts wearing down, those iron boots are going to be a liability, rather than a boost. To use them to their fullest you'll need a full iron suit.