I drink water at work from a plastic bottle, refilled at cooler. I wonder if putting lemon cuts (0.25 lemon/L) in it will preserve it and decent way to keep water safe. Is there a chance that lemon can go bad if left in bottle for two or three days?
This is more likely to introduce organic matter (the lemon cuts) that can induce spoilage than prevent it... common experience shows that food that has "some lemon juice" added (and the amount you are suggesting is in the same order of magnitude) is not significantly preserved by that, so we can assume the acid (which would be the preserving factor) is far too diluted in that mixture to keep either the water or the lemons from spoiling. Also, lemons as bought from a store or vegetable stand are far too widely variable in their acid content to make any statement on chemical potency - the only remotely safe way to preserve anything via the acid from natural produce would be to educate yourself on the suitable pH range, then after each preparation take a sample and examine it with a pH measuring device (pH meter, indicating paper - do not dip such devices directly into anything that still gets served as food or drink!).