I use reaper mostly on windows - mostly due to 3rd party plugins and it is the only time I use window - but occasionally also use Reaper on my main computer, which is running Ubuntu Linux.
I have found Reaper on Linux (the native version) to be running very smoothly and to be very stable indeed. All bundled plugins that come with Reaper work, as far as i can tell, on Linux as well. The only problem, as you suspect, are 3rd party plugins (i.e. VSTs) and also the SWS extensions which do not seem to be currently available on Linux. It is possible to run some VSTs on Reaper/Linux using wine, but it was not the smoothest experience for me.
I have used reaper on Linux mainly to practice mixing wav stems (exported from my windows machine) using reaper's bundled effects (compressor, reverb etc, even a simple guitar amp emulator) and it never ever crashed on me. In general I find Reaper an amazing program, and it's smooth Linux implementation (and the fact it exists!) really sealed the deal for me.
I would recommend downloading a fully functional trial version from http://reaper.fm/download.php (don't be put off by the fact that the Linux version is experimental, it is very stable) and go from there.
Reaper also is extremely reasonably priced and the license covers both your Linux and Windows machine - as long as you only run Reaper on one machine at a time.