Storing equipment in shed?
Changes in conditions can kill equipment in environments like you're describing - especially involving high humidity. When the temperature falls at night, the moisture condenses on every available surface. The next day as the air heats back up, the moisture evaporates. If the day/night temperature swings are big enough to cause significant condensation, then you're essentially dousing any metal surfaces in water every 24 hours. I had a home years ago with a poorly built uninsulated garage, on a cool summer morning there would be significant condensation - any tools left out there would look like they'd been sprayed down by a hose. Sometimes the condensation would be so great that the tin roof would drip water and it would "rain" inside the garage. It killed a few tools. I quickly learned to deal with it (and made structural improvements) and things were fine - I still have and use some tools that had been stored in that garage for a decade, once I learned to protect them.

There are ways to combat this. Wipe down metal surfaces with a rag and some light machine oil every time you use the tools. The oil will help repel condensation. If you can, store the tools in tightly-closing cases or bags, and then put the cases in airtight storage bins. This will cut down on the condensation significantly by capturing a more-stable environment directly around the tools. For the cordless tools, it probably makes sense to store the batteries in your house regardless of the temperature or humidity, so they can be on or near their chargers (I'm assuming you don't have power at the shed).

You said there is no floor to the shed - if the floor is earth or gravel, it can be a big source of moisture. For the larger items like the lawnmower, it may make sense to store them on a pallet or at least on a makeshift floor of 2x4s or other scrap lumber - the idea is to allow air circulation under and around the tools, so they're not just sitting directly on the damp ground all the time.