Does installing and using Wine open up your Linux platform to Windows viruses?
Yes and no.
Viruses/trojans are just programs, and will work on Wine... Also, your normal Linux file system is exposed to Wine with the user that launches Wine credentials.
BUT, usually viruses are based on lots of hacks, and they expect a "standard" and common Windows installation. I doubt that any virus is coded thinking that it will be executed on Wine, and if it exists, it will probably not be too successful.
Why? Because Wine users are a small portion of normal users, they have "weird" and strange installations (think in all flavours of Linux+Wine), usually are avanced users, and they have a strong community aware of security.
So: yes, you are exposed to windows viruses, but not totally exposed, and most probably your linux installation will not be contaminated. Just be careful as you are on windows.
On the other hand, you can use several techniques to increase the security: use chrooted wine (search google for chroot), virtualized environments, etc...
Yes and no. Some viruses will work on wine particulary those which uses programs security bugs you are running under wine (wine running old version of adobe flash will not automagically protect you from exploiting it). Also the Linux FS is exposed in default configuration.
On the other hand viruses may depend on particular configuration and if it tries to exploit Windows kernel - it is bad day for him. Also emulating IE by gecko may not be helpful for exploits.
At the end - the fact that virus have affected wine does not mean it gain access to all files. It usually is not aware that Z: is director where you keep your files so it may simply ignore it. Deleting virus may be as simple as removing
~/.wine and it will not delete most of your settings.
There is an old Linux.com article, in which trying to infect a Linux install with Wine is actually tested. If you are of the TL;DNR type: it doesn't work. You are fine without antivirus.