Are there any safe places to store my unneeded weapons and shields?
Solution 1:
Yes
Once you purchase the house in Hateno Village for 3,000 rupies and 30 bundles of wood, you will get one weapon rack in that house which allows to store one weapon in it. Though I haven't tested it in the long run I'm sure that the weapon you put in will not disappear when you come back.
You can then buy more furniture from the same person you bought the house from, allowing you to get more weapon racks among other things.