So in this case, the issue is not duplicated vertices, but rather disconnected faces. If you don't mind watching me ramble and make mistakes for nearly 38 minutes, I made a video for you showing how even fixing the issue of disconnected faces does not completely solve the issue with the inset tool you were having, because the corner faces will very quickly overlap each other and cause problems.
Basically you need to approach the modelling differently, as demonstrated in the video. There are many ways to do this. I showed a few ways in the video, but the best way (near the end of the video) in my opinion is just to extrude the perimeter walls up, extrude in, fix any edges that went the wrong way, then extrude up again. At the very least you might learn a few new things about Blender from watching me goof up in the video. PS. Sorry the audio is quite low, you'll probably have to turn the volume up. :-)
Also, YouTube is still processing the higher res videos, so you will have to check back in a little while to watch the full quality version (1440p): https://youtu.be/cci59mn9dcw