Yes, wall mounted. I found the organizers on Amazon, see here. The testers are useful. I've had no problems with batteries falling out, and this is in a room with a heavy door to the garage that gets opened and closed daily. No lids on these units.
This depends a lot on the specific charger.
Some chargers will trickle charge the battery forever until you remove it. Some will have a very tiny current leak that slowly discharges the battery. Some go total open circuit so the battery is essentially disconnected.
I can say you probably DON'T want to leave the battery in the charger if it's unplugged. That is far more likely to drain the cells- charger circuitry that's designed to charge the cell when backed with wall power could discharge the cell by providing a slow current path for battery power to leak away.
What you should do, IMHO, is get a battery storage rack like this one. Search Amazon for 'battery rack' and you'll find several similar items. Once your batteries are done charging, take them off the charger and put them in that rack. This is especially helpful as you transition more of your devices to rechargeables- buy rechargeable batteries in 8 or 16 packs, charge them up (remember Eneloops are only factory charged to 60-80%) and put them on that rack. Then you can do a first-in-first-out system- start at the beginning of the AA area in the rack, now put new charged ones after the last one, but take ones out from the beginning.
And if I ever get AAA’s, are these chargers brand agnostic? Could I get ludda browns from IKEA, for example? Or must be eneloops? UPDATE: I was doing some reading and I guess ludda/eneloop are the same battery, different brand! How about other batteries that aren’t eneloop/ludda, like energizer? Is the charger agnostic to the brand?
Eneloops are high quality NiMH (Nickel Metal Hydride) batteries. NiMH cells need a specific charging strategy- you can't just shove power into the cell; figuring out when to STOP charging the battery is important so you don't overcharge and damage the cell. A NiMH charger uses a thing called -dV/dT charging. That means negative delta voltage / delta temperature. When you charge a NiMH cell, as you put in more power the cell voltage increases. But when the cell becomes fully charged, the voltage actually goes DOWN- the cell is converting more of that energy into heat. So once you see a sharp spike in temperature and/or voltage is starting to go down rather than up, you know the charge is done and you either stop charging or switch to trickle charge.
This same strategy will work on any NiMH cell, brand isn't important. So you can get NiMH cells from Ikea or any other supplier and charge them in your Eneloop charger / charge Eneloops in your other-brand charger. You want to make sure you are using a good quality charger, that has good -dV/dT strategy. [This site](https://lygte-info.dk/info/indexBatteriesAndChargers%20UK.html] has lots of good reviews. However suffice it to say, the Eneloop branded 4 slot charger is generally considered to be a good choice.
As for AAAs- Eneloop AAAs are still NiMH cells. The only limitation with them is they're physically smaller, so you can't charge them at as high an amperage as AAs. With the Eneloop charger this is no worry, just throw the AAAs in there. If you use a 3rd party faster / higher output charger, you might want to turn the output down to 500mA or below when charging AAAs.
HOWEVER, if you use ANY other battery chemistry (LI-ION, LI-Polymer, LiFePO4, NiCad, Alkaline) do NOT use your Eneloop charger to charge them. Multi-chargers DO exist- I personally run a Nitecore d4 charger which charges my Eneloops and also lithium cells like 18650's for flashlights (I recommend this unit).
Hope that helps!