This is an interesting question. You can absolutely put multiple operating systems on a Pi3 although I have never done this because I have so many SD cards. Assuming your media for RasPlex is on some kind of NAS storage and not on the card it should be fine. Most distros install on about 4GB and run nicely with 8GB on the partition.
Regarding which distro to pick, there are so many to choose from. My favorite lite desktop is Ubuntu Mate. Although they are small, I don't believe the other lightweight distros like Arch and RaspBSD support browsers at this time. Gentoo might be an option as well but have never used it. If you are using one of the standard offerings like Ubuntu Mate, you can set up a dual boot using NOOBS
The lightest thing possible may be to use a distro called PiNet (you can learn more about this great project here) but this would be truly experimental IMO. I have had some very limited success getting it to work with videos, picamera and minecraft, but if you have another laptop or Desktop system on your network and you were up to the challenge, this could be an interesting project to try.
PiNet is a distro designed for classroom setups and uses LTSP to set up a thin client on the Pi and pxe boots to mount the OS from the network. This is probably the lightest thing you can get since the OS is not actually on the SD card but on the networked computer instead. The draw back is that you are technically remotely logged into the terminal server so there is some setup involved with making it work and performance may depend on the particulars of your network. If you want to give it a shot this is a link to setting up RealVNC to remotely take advantage of accelerated graphics for viewing videos which you would need to use OMX Player
