Although it is not phrased explicitly as a "limitation", I think it is reasonable that the fatigue penalties are waived for sleeping in medium Mithral armour.
A few class abilities only work if character is wearing Light or no armour. These seem to be clearly "limitations" and definitely waived for a Mithral Breastplate.
I have generally counted the fatigue rule for sleeping in armour along with the skill and class abilities that are only allowed in Light or no armour.
It is a rare situational bonus anyway - an ambush against sleeping PCs, and preventing their use of armour might be a possibility, but if it is a common threat in an adventure, the players will figure out ways to to avoid it as much as they can.
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
Well, you have the breastplate and the effects of mithral, which you combine to produce a final mithral breastplate:
\begin{array}{l c} \text{Armor} & \text{Type} & \text{Cost} & \text{AC} & \text{Max Dex} & \text{ACP} & \text{ASF} & \text{Movement} & \text{Weight} \\ \hline \text{Breastplate} & \text{Medium} & 200\text{ gp} & +6 & +3 & -4 & 25\% & 20\text{ ft.}, 15\text{ ft.} & 30\text{ lbs.} \\ & & & & plus \\ \text{Material} & \text{Type} & \text{Cost} & \text{AC} & \text{Max Dex} & \text{ACP} & \text{ASF} & \text{Movement} & \text{Weight} \\ \hline \text{Mithral} & -1 & 4,000\text{ gp} & 0 & +2 & +3 & -10\% & \text{(varies)} & \text{(half)} \\ & & & & equals \\ \text{Final Item} & \text{Type} & \text{Cost} & \text{AC} & \text{Max Dex} & \text{ACP} & \text{ASF} & \text{Movement} & \text{Weight} \\ \hline \text{Mithral} \\ \text{Breastplate} & \text{Light}\dagger & 4,200\text{ gp}\ddagger & +6 & +5 & -1\ddagger & 15\% & 30\text{ ft.}, 20\text{ ft.} & 15\text{ lbs.} \\ \end{array}
Unfortunately, they don’t tend to put special materials in neat tables like the one I just made for Mithral, so you have to read carefully to determine its effects, but generally speaking the process is the same as what I’ve done here: you figure out what the material modifies, and just apply each one.
\$\dagger\$ In Pathfinder, mithral does not change the armor’s proficiency requirements, so even though the mithral breastplate is a light armor, you still need proficiency in medium armors to use it properly. This is a new rule in Pathfinder that did not exist in 3.5, and I recommend ignoring it.
\$\ddagger\$ Note that mithral items are always masterwork and the price of that is included in the 4,000 gp, so you don’t have to pay for it separately (i.e. another 150 gp).
About Armor Check Penalty, the rules are less clear: they never explicitly say that the Armor Check Penalty reduction of the mithral includes the Armor Check Penalty reduction of being masterwork armor. The rules for adamantine armor are a little more clear on this point, and the formatting of each material should be consistent, so that’s a clue, but not explicit. Consulting with various example mithral armors,1 it appears that it is included (i.e. the total reduction in Armor Check Penalty is −3, not −3 for mithral and −1 for masterwork) is another clue, though that’s not necessarily useful since the mithral shirt also weighs less than it should (10 lbs instead of 12½ lbs).
So by strict RAW, this isn’t exactly definitive; the primary source on this is the rules text for mithral itself, which is ambiguous. You could make an argument that the masterwork effect is not included, and then that the example armors are mistaken or intentionally different from what the usual rules would provide, and also that nothing says that adamantine and mithral have to handle this the same way. But that’s a lot to claim, and since the rules for mithral are merely ambiguous, rather explicitly in your favor, I’d be shocked if any DM bought it, but the argument could be made and I might buy it for a TO exercise or something. But then I’d also consider houseruling that you get both (i.e. ACP 0) just because I might be convinced things are better that way.