• probably not, Linux isn't running when you're in Windows, and Windows isn't running when you're in Linux
• it could, but I think you'd need a targeted attack for malware to jump from Windows to Linux, since that's a pretty niche target
• yes, if it has write access, you're open to ransomware attacks, which are a fairly common form of malware; if your NAS has a rollback option, you're probably fine, but definitely make sure your remote backup restore works (you do have off-site backups, right?)

If you want to be extra secure, encrypt your Linux partition. They could still corrupt your Linux partition, but they wouldn't be able to read anything on it without your password. Both of my Linux machines (laptop and desktop) use an encrypted root partition, and they run games and whatnot just fine (I don't notice a slowdown).