Personally, I don't recommend using Windows for production servers, since Windows is nothing but one security patch after another. Whatever time you save from alleged "easy installation" is more than made up by the requirement of continually riding herd on the OS. But that's my own silly little prejudice, and I realize that the number of production Windows appservers is not likely to plummet anytime soon.
If installing Tomcat in Linux seems difficult, that's just because you're not as comfortable with Linux as with Windows. As far as it goes, I'd have to go back and RTFM to get Windows to run Tomcat as a service, but I run multiple Tomcat servers without a second thought. It's simply a matter of what you're used to.
MySQL is slightly more comfortable on Linux boxes, since that's its background. Then again, SQL Server doesn't run (native) on Linux at all. Such is life.
But a WAR is a WAR, and
Java is "write once, run anywhere". I've developed webapps for years using a Windows XP desktop and Linux and Solaris production servers. As long as you don't hard-code OS-specific things (like references to "C:\Program Files"), the exact same WAR file can run on any of those platforms.
The secret of how to be miserable is to constantly expect things are going to happen the way that they are "supposed" to happen.
You can have faith, which carries the understanding that you may be disappointed. Then there's being a willfully-blind idiot, which virtually guarantees it.