Yes, of course. This is the big advantage of pico]OS, because it comes with the BSD license. As long as you do not remove the copyright notice in the source files, you are free to copy, modify and use the files. You can use pico]OS in conjunction with other types of open source, as long as the pico]OS license does not conflict with other licenses. Please see the pico]OS license for details.
I strongly recommend you to use the makefiles. Some preprocessor defines are passed to the compiler that could be lost if you will try to compile the RTOS from within an IDE. So I recommend you to compile at least the pico]OS library with the provided makefiles. The library can then be added to your project in the IDE. Also, most text editors allow to call external makefiles to build user specific projects.