Armor proficiency is more want than need.
Animals are normally not proficient with armor. As per PRD Armor Proficiency feat:
Normal: A character who is wearing armor with which he is not proficient applies its armor check penalty to attack rolls and to all skill checks that involve moving.
Usually only light armor (leather or masterwork studded leather, mithral kikko) is worn with no proficiency, but it is up to you to decide if the benefit of having more armor outweights the cost of armor check penalties. For example, a Mithral Full Plate has armor bonus +9 and armor check penalty -3. In Pathfinder, you can spend three feats to get Light Armor Proficiency, Medium Armor Proficiency and then Heavy Armor Proficiency to limit applicability of this penalty, but you do not have to.
Masterwork cost is added after everything else.
So, for a large quadruped, the cost of armor is: final_cost = base_cost * 4 + 150gp (this gets a bit tricky when you consider special materials, which have masterwork cost built-in, but reverse-engineering works).
The rules for multiplying costs and weight to get the additional cost of mithral apply only to weapons and non-armor items.
It is perfectly reasonable to say that the price of raw mithral stays the same regardless of it's intended use. That said, the price of various mithral items is influenced by other factors, not only the price of raw material, which is why the disparate pricing seems to occur. Looking at it the other way, you would never use 50 lbs of mithril to make a 25 lbs of resulting armor. Maybe some small amount on top of 25 lbs would get wasted, which is unlikely as usually metals are easily recycled. Even so, the weight of a mithril armor equals to not only weight of mithril plates themselves, but also leather straps, various kinds of fastenings such as wires or chains, gap-oprotecting chainmails, under-armor padding etc.
To complete your example, I would say that for an additional cost of 9000 gp you get 18 lbs of raw mithral, slightly less when you consider a masterworking fee of 150 gp included in the price. Subtracting the 18 lbs from 25 lbs of total armor weight, we are left with 7 lbs. It seems reasonable that for a full plate weighing 50 lbs, 7 lbs would be accounted to padding, fastenings, decoration etc. which is not lightened by using mithral. The price of 500 gp/lb holds.
Now moving on to weapons and other non-armor items, the cost of working mithral is calculated by taking the original weight. DuckTapeAl already established that mithral is a defensive material, so maybe it is the case that effective price doubling is related to additional effort put into shaping, hardening and sharpening a slab of mithral into a weapon. Also, with weapons proper balancing is paramount, so while making a sword from mithral, you would actually have to use lighter (and probably more expensive) hilt material, additional weights (preferably also mithral) to keep it up to speed. It's even worse with blunt weapons, as a warhammer needs to be tip-heavy for sufficient momentum, forcing the smith to re-engineer hilt diameter, length and weight to account for the lighter head. On top of that mithral is traditionally said as being very hard to work, which would force the smith to use more sophisticated tools and materials, such as better furnace fuel (for more heat), different hammers (that would not break when striking), unusual tempering fluids (gentler or more volatile) and more resistant sharpening stones (perhaps diamond dust or something similar). All in all that could possibly double the price to procure a specific weapon, even if indeed you need less raw mithral, again making the 500 gp/lbs rule hold.
Best Answer
Either, depending on how you want to purchase the item. Mithral Large Barding (it's large barding, only mithral!) is 15000 gp, while Large Mithral Barding (it's mithral barding, only large!)is 42000 gp. Their stats are otherwise identical. As a player, you should generally choose the best of a set of actions which are nonmechanically equivalent, so Mithral Large Barding is the way to go. In campaigns I run, the pointlessly more expensive versions of items are not generally available, unless I want to make a point about a region being mechanically backwards.
Thorough explanation for you confused peoples:
We start with an item, in this case Full Plate. When we consider Full Plate, there are lots of possible kinds of Full Plate to buy. We could buy spiked full plate. We could buy Large full plate. We could buy unusually shaped full plate. etc.
For the purposes of this explanation, we note that both mithral full plate and Large full plate are available for purchase, at different prices and with different stats.
Next we look mithral or Large full plate. Large full plate is still an item primarily composed of metal, and so it can benefit from being made of mithral, changing it's price and stats. Mithral full plate is still sized for a Medium creature, and so is eligible to instead be sized for a larger creature, changing its price and stats. The end result is that the stats of the two armor sets are identical, with the exception of cost.
The rules for modification of items always assume a "base item" that the augmented item is "similar to, except...". This is why the prices for such things are expressed as operators (+500, X3,-10%, etc.) rather than numbers. This presents a fundamental challenge for the application of multiple modifiers, which Pathfinder deals with by not allowing multiple modifications. All pathfinder mods of this nature apply their effects to a base item and result in a new item, which can then be further modified by using it as the input base item into another modifier. You can't apply these affects simultaneously, or both to some 'original' base item; that's not how the system works. Instead you have to calculate one modification and then use that as the base item for the other.