This is going to be pretty dependent on where you're driving and in what conditions. If you're going to encounter a lot of snow, you'll first want to measure the tread depth and make sure you're at
5/32nds of an inch, at least. It looks like you probably are, but worth checking.
I'd also recommend, at a minimum, having two of the same tire on each axel. So if you can directionally do two Blizzaks in the back, you'll want to go that route. (Better/newer tires go on the back.) If you have to mix them on each axel, I'd consider replacing some/all. That said, it is highly recommended to have matching front/rears when it comes to winter tires.
Two articles worth checking out:
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Dangers of Mixing Tread Depth
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Mismatching
Basically the problem you're going to have is that the Blizzaks are a great tire, and the off-brands are, well, not. This creates the potential to have significant traction differences at each end of your car, putting you at a big disadvantage. So again, as a minimum, put the Blizzaks in the back.
As far as the clear coat issues go, I'd probably throw some plastidip on there and call it a day. Should help protect it from further corrosion, etc.
Hope this helps - congrats on the new car!