Fixing Punctures / Puncture Repair / Exploding Tubes

Rim tape - good idea. I'll check that and then start swapping tyres.

But what about the fact that I can see no bubbles in the tube when I put it under water? Is it plausible for a puncture to be that slow?

Very easy to have such a slow leak you can't see bubbles. If you are only loosing 30 PSI over night then that only a small volume of air escaping over a long time. Also the tire is pumped up to a relatively low pressure when out of the tyre and your checking it in the water so the rate of leak will be even less.