1) Check your local Craigslist for winter tires - I got basically brand-new Blizzaks for both me and my wife for $300 a SET from a couple random people. Also, as much as I disdain Facebook, they have some great groups for people buying/selling/trading rims/wheels in your local areas that are a wealth of opportunity.

2) You CAN put 16" rims on these cars, but very few (if any) of us have actually done it. You're going to be looking at getting tires in the range of 205/65/R16 to get close to OEM size so your speedo isn't way off. Use the Tire size calc in my signature for more info

3) I got a set of 17" rims off a guy on FB then, like I mentioned, the blizzaks from CL and only spent $650 total including mount/balance. This will be my 4th (and final) season with them as they're showing their age.

Pic for reference:

 

I don't know, unless you are going through deep snow I just don't see the benefit from a skinnier tire. I use 15" on my car because they are on hand and I don't have the cash to switch them. I would run the same size as my summers if I could afford them. You sacrifice handling by a large margin, in my opinion, with tall skinny tires. It is a very noticeable difference. When my current set wears out I will be going larger.

Also, even some basic cars are coming with 17" tires so the new "norm" is very likely a 17" tire and they may be more plentiful and cheaper.