Quick answers: No, no, and maybe.
It appears that you may be confusing the TWRP app (which is basically useless bloat) with the actual TWRP recovery. You need to find and download the latest TWRP recovery for your device model. If you can find a ZIP of it, great; because CWM can then install TWRP recovery for you (so that when you reboot into recovery, you'll see TWRP instead of CWM). In my case, there was no ZIP available for the latest TWRP; it was available only as an IMG file. An old version of TWRP recovery, however, was and likely still is available in a ZIP file. So use CWM to install that ZIP of older TWRP, which can then install newer TWRP as an IMG file. I'm just a struggling noob myself, but I'm pretty sure CWM is not able to install an IMG file. Good luck!