If you wish, you can put your alkaline batteries in the fridge or freezer, but they have around a 7 year shelf life so this really isn't necessary.
Rechargeable batteries should be fully charged, then stored at room temperature. They should be discharged and recharged at least once a year if you are storing them for this long. But if you are storing any battery for over a year you may be buying too many at a time.
Fire shouldn't be an issue if you are storing them in a fashion that prevents them from shorting out. The easy way to do this is to put them back into the original package when storing them. You can also put them into individual plastic bags. One battery to a bag will prevent any shorts.
While local codes may vary, Federal codes permit the disposal of alkaline, Li-ion, and NiMH batteries in your local municipal trash stream.
NiCad batteries need to be recycled as the cadmium in them is a toxic heavy metal. Most places that sell NiCad batteries will also recycle them. This is free and it doesn't matter if you bought the battery there. They are required to recycle any NiCad brought in.
