Mixing different brands are usually acceptable, while mixing summer and all season changes the gripping characteristics and can be dangerous.

Always put the best [Michelin] on the rear at least you won't have to worry about sudden power on wet oversteer. Weaker and less gripy [in cold] on front will just increase understeer.

Using different tread patterns on front and back shouldn't be an issue. Even among the same tire pattern you will have different grip characteristics when turning as the rear are a different size and different camber. Even among some tire models sometimes you will have different tire tread patterns from one year to the next. So if you bought a tire last year, it may not have the identical tread pattern this year.

The real concern would be having two different tread patterns at the same axle (as reference). That would make thing more hairy and would really mess up the tires.

The difference in compound, design, how the operate at different ambient temps, internal construction, etc etc makes mixing those a really bad idea. You can read more about that here,

Mixing Tires