How can I use two different IDEs in two different operating systems with the same project?
Should I use a single cross-platform IDE?
No. I think this is a question of the wrong question. To build a cross-platform project, your build scripts and the system-neutral nature of your code are important. Sometimes this can help create project files for your preferred development environment, but supporting multiple project files for multiple IDEs will only make it harder and harder for you. Instead, you should focus on finding a build system that minimizes the time spent on project maintenance.
For this, CMake and PreMake seem to be two of the best tools that can happen.
There are dozens of alternatives (such as SCons, Cook, kbuild, Jam and Boost Jam, and many others), but since CMake and PreMake generate project files and create scripts, they may be better solutions.
Your mileage will be different.
Could / Should I post my code on the Internet and automatically synchronize it with stand-alone projects?
You must have a reliable source of control that works wherever you are. Git and Mercurial seem to work best if you use some kind of "cloud" hosting, such as Github or BitBucket, but they by no means require it. Depending on your work environment and team size, you may prefer Subversion or PerForce or something else, but it depends on you and your team.
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