Can natural gas go stale in the pipe

This does NOT sound at all like a normal situation. Either your gas fireplace has a serious flaw or the unit is vented incorrectly. I would stop lighting it and immediately get a professional in to look at it. At the same time you should use every avenue at your disposal to get the builder, contractor, and gas fireplace installer on the hook to look at this unit and replace it if necessary with a properly functioning and correctly installed unit.

The click-click-click sound is probably coming from the water meter.

Check if it makes the same sound when you run COLD water, in a kitchen sink, for example.

Sometimes a mixing valve is used to put some hot and some cold water into the toilet tank to prevent or limit the tank sweating in the summertime. The mixing valve is adjusted to "temper" the water by adding just enough hot water to raise the temperature of the water in the tank from cold, to I'd say "lukewarm", but not hot. The water is then much closer to room temperature, so it won't "feel" hot, or even particularly warm. If a mixing valve is installed and the water feels hot or even particularly warm, or even on the cold side, the valve needs to be adjusted.

In either case, cold water only, or a mixture of hot and cold, some water meters just "click" as they measure the amount of water flowing through it.