Maybe - really depends on what you feel most comfortable with. As the project is non-graphical, all the IDE gives you is editing of files and compilation. So you can build the project on one machine with the IDE there, and then move the sources to another machine for compiling there.
I personally would just have two makefiles, one for Linux and one for Widnows. Makes life fairly simple [you could have a "outer" makefile that picks the right one based on some clever method].
Yes, you should find a version control system that works for both Windows and Linux (git, mercurial, subversion, bazaar and several others). That way, not only do you have a central repository [you can use either of your machines as "server" for any of these], but it also allows you to keep track of your changes. Definitely worthwile doing!
There are hundreds of different alternatives. But the simpler you keep it, and the less complicated your tools are, the more time you get to spend on actually programming your project, rather than, for example, figure out why CMake isn't building the way you want it to.
Also, make sure that you separate out all your system-specific code to one file per architecture. That way, it's easy to port to another architecture later, and it makes MOST of your code compile on both systems.