bernardyeo,
Yes, all Evos and for that matter all 4WD vehicles will have a centre diff.
The problem with using different tyre brands or even models on the front and rear axle is that they will have different wear rates; the variance is subtle but you can be sure that your differential will know it.
This doesn't mean that the diff will surely die, but it introduces unnecessary strain on it for no apparent gain. If you're using different compounds on different axles to adjust the vehicle's balance, then fair enough.
Soundstream
Stock alignment settings depends on which Evo, as they tend to change this on a car by car basis. Generally, you'll have more negative camber (-1.5) at the front and slightly less at the back (-1.0) with zero toe front and rear. This is a simple setup that will give you decent cornering grip with decent tyre wear.
If you want better turn-in, add a couple mm of toe out in front, though this will cause the nose to wander at high speeds and will also increase tyre wear.
For the rear, leave it at 0 toe, unless your AYC is kicking the rear end out under power, in which case you should toe-in the rear to reduce the severity of the AYC's intervention.
Alignment is quite specific to how you want to drive, so it's better to find out how changes in alignment affect the car's dynamics and then develop a setup that complements your style rather than just going stock. A tailored setup and a stock setup will cost you the same, so may as well get something that is better for you.
As for suspension, are you running adjustable coilovers or standard suspension? Adjustment, again, depends on what you want to do with the car and what you want the car to stop doing.
Also, it's not advisable to adjust the suspension unless you're 100% certain it's the problem. 9 out of 10 times, handling issues are usually down to driving style.