You didn't get what I said at all...yes I want it to be faster I only listed what I thought it would take to make it faster as a minimum guideline. It doesn't take 200BHP to make a 2800lbs car go faster than a 110BHP 2200lb car.
The AE86 is considered slow now, whether you like it or not. It may have been fast (or at least reasonable) in it's day, but compared to a lot of cars nowadays it would be considered slow. If you want the new iteration of it to be slow as well, by all means ask for only 140hp for 2600lbs. I personally don't want it to be 'only a little bit faster' than the 20-year-old AE86, I want it to be new, I want it to feel like it's a sports car, not something my mum's Ford Fiesta could do (example, my mum does not actually own a Fiesta 
)
LoL, you don't get why Mazda built the Renesis yet you used it as an example, and those are
my points in bold.
Exactly, I don't see why mazda built it... I can see very little it has over the RX7 - looks worse (IMO), less power, slower acceleration, heavier, etc.
Increasing the displacement or even bolting on a turbocharger when they could increase the power with better technology is not advancement it is called "Progress" a word applied many times to things that do not signify advancement. It has also been considered a technological advancement by auto journalists but that doesn't make it true.
Is that not what Toyota is trying to do with things like VVt-i, valvematic, etc.? And the same thing turbo manufacturers do by researching and bringing out types of turbo's that spool faster, or make more boost faster?
The Mazda Rotary HP did peak when they turbocharged it but it was for a $30k car over 10 years ago. The RX-8 motor is naturally aspirated, lighter, and almost matches the HP but it takes RPM to do what it does and sells for the same $30k base price 10 years later. Success it is/was. It also handles better I hear.
As you said, you don't need a heap of power to make a car faster (in fact, that's not what you want), but Mazda screwed it up by making the RX8 both heavier and making it less powerful.
"The whole point is to make a car faster, and more efficient at the same time, while retaining or adding extra amenities."
The whole point of what? Many cars are faster than they used to be and they are not even close to as fuel efficient as they were. You can blame safety regulations or you can blame emissions, but that doesn't change the facts that most cars are simply built to be larger than they used to be and no advantage is gained other than space. If we wanted larger cars we would buy them right?
Cars are more efficient now than they ever were. 
Both only a few years back, and with differences of ~15 years
I am just suggesting that it doesn't take a bigger engine to make the car faster than the old one when the only difference is more weight. That's what engine technology is good at doing, worse case there is always turbo/supercharging.
I agree, but the safety regulations prevent this, cars are required to have more airbags, crumple zones, etc. than ever before, which increases weight and then you have to increse power.
The heavier Subaru engined car isn't going to be more fuel efficient than the original Hachi-Roku's, not by a long shot. And the engine will not just be more powerful but it will also be larger and yet still lighter(aluminum block). For a $20k entry level RWD car you can't ask for everything and get it.
I'm not sure about that, as an example I can use the 2010 Nissan Maxima vs. the 1994 Nissan Maxima, although the 2010 model has a larger engine(3.6L vs. 3L), it is far more efficient at 22MPG vs the 1994's 19MPG.
Ferrari is a very bad example, yes they are a "Progressive" company in regards to HP. But they have to resort to hybrid technology like Electric motors and batteries to keep cylinder count up due to
Gov't regulations. No applicable points to be made there.
Ferrai use no Hybrid technology in the 458, the 458 uses a 4.5L engine compared to the 430's 4.3L, but manages to make an extra ~60hp.
A better example of the effect is the bigger block engines of the US 60's. Engine power was increasing with the OHV engine becoming mainstream after the "Rocket" was introduced in the 50's but after the technology stopped paying off as quickly they resorted to increasing the bores/stroke to keep increasing power. Eventually they had to use leaded gasoline and high compression to get more HP from the then huge displacement engines. Good times those 60s were Gas was cheap
and emissions regulations were low. But the increase in power wasn't just technology it was increases in displacement and used higher octane leaded gasoline to keep increasing the compression ratio. And that's my point.
Yes, I understand that, and I agree there is a time when you get too much displacement, or too large a turbo, and it becomes excessive, which is why manufacturers should continue devloping technologies to pull more power out from every drop of fuel. There is still a very long way to go... You know all the heat that gets transferred to the engine cooling system (radiator), all of that heat is extra energy coming out of the fuel and not being harnessed by the engine, if they keep improving technologies this should eventually become near-perfect, which is when the manufacturers will need to begin to up boost levels or displacement to get more power.