I tried it with getting exactly 4 perfects from Chun Li (in her stage), Boxer, Claw and Dhalsim (being the last of the eight in the boss rush) and I fought Akuma. Therefore the last of the boss rush fights indeed counts and so it should be possible to fight both Sagat and Akuma with only 7 perfects altogether.
But Akuma really felt somewhat easier in this version. Did they really nerf him (as someone on YouTube told)?
Okay, summarizing in order to fight Sagat and Akuma you have to get 4 perfects from the following fights:
Sagat: - Balrog - Vega - the boss rush battles
Akuma: - the initial 8 - Balrog - Vega - the final one in the boss rush - Sagat
Therefore, it is possible to fight both with only 5 perfects altogether. I started right at the final stages and tested whether you still can get Akuma and it's indeed possible because Sagat does count towards Akuma. You can get the perfect, then lose to Bison and then beat them both without a perfect and still get Akuma. However, I didn't test what happens if you use a continue after getting the perfect from Sagat.
And did I just beat Sagat, Bison and Akuma with a single life only using one E-tank right before finishing Sagat...?
Hmm, so if you can get a perfect against Sagat, lose to Bison, and then non-perfect Sagat and the first one still counts toward Akuma, I wonder if you could get multiple Perfects off of Sagat by losing to Bison repeatedly.
You can continue as much as you want if you want to fight Akuma. Even if you lose to Bison after perfecting Sagat, and then use a continue, the aforementioned perfect still counts. However, against Sagat you cannot use continues as soon as you obtained your first perfect towards him (but you can use continues before this).
Also notable is that if you get a perfect from Bison while having 3 perfects towards Akuma, you will not fight Akuma, but you will get the "better ending". After the credits I got the cheat code for Guile's theme. IIRC you only are supposed to get that after beating Akuma, aren't you?
So the only remaining question I can think of right now is WazzupGenius's?