Different tire sizes, tire brands, and tire styles should never be mixed on the same axle. Tires of different sizes can be mounted on the front and rear axles if approved by the vehicle manufacturer. It is common on high performance sports cars and luxury sedans for the rear tires to be wider than the front tires.
If your vehicle does come equipped with a different size on the rear, you should keep the same size combination front to rear when it comes time for replacement tires. In this situation, it is also okay to replace 2 tires at once, but it is recommended to be the same brand and model as the tires paired on the other axle. The reason for not mixing brands and tire styles is that each tire brand and even tire model has different performance attributes. The most predictable handling is when all four tires are the same brand and model, and aligned with the size recommendations of the vehicle manufacturer.
The mixing of winter, summer or all-season run-flat tires with non-run-flat tires on the same vehicle is not recommended under any circumstances. Run-flat tires handle very differently to standard tires, and mixing them could cause very unpredictable handling and other safety issues.
A pair of winter tires can be mixed with all-season or summer tires, but they should always be installed on the rear of the vehicle. Ideally four winter tires should be installed during the winter months in northern climates.
Finally, you want to rotate your tires every 5,000 miles so all four tires on your vehicle wear out at the same time. This allows you to purchase 4 new tires for optimal performance and safety.