Water – Can’t get NPT connection to not leak

There is nothing about this condition that by itself suggests a leak. Common causes are your cold water pipes run near heating ducts or the furnace, or water from your hot water heater expanding back into the cold supply.

Likewise, a warm spot on your floor could be from ducting, a light below, the sun hitting it through a window, etc.

If there is a leak, sooner or later you will see signs of water - drips, stains on the drywall. If you do don't see any of these, I wouldn't worry.

With out a more detailed description of the pipe it will be difficult to answer precisely, but here is a general, "good for most" answer. Since you can detect a sewer odor you can assume the pipe is either a vent or a drain for the kitchen. I'll also assume that it is a plastic (ABS) pipe as you stated it broke after it was pulled on (unless it is very weak iron?). Either way the repair is straight forward. You must cut the damaged section out so that when removed you will have sound sections of pipe to fit the repair too. It is probably 2 inch diameter ABS. Purchase (2) rubber pipe connectors (they should have hose clamps on them) and a length of pipe (ABS) the size of the damaged pipe. Slip one coupler on the new section and the other to the existing pipe (use dish soap as a lubricant). hand tighten and check for drips.