Why do my hotkeys lose their bindings?

I am currently binding number key 1 to an Iron Sword, and number key 2 to a bow. However, when I press #2 to use the Bow, and press #1 to use the Iron Sword again, the binding to bring up the bow disappears.

How can I quickly switch between bow and sword/shield?

The number hotkeys do not work well if you favorite a weapon or shield that becomes part of a stack (as mentionned by yx). Early on in the game this happens a lot as you use regular items, but is less of an issue as you tend to get magical weapons with unique effects that will rarely match exactly that dropped by the enemy.

Pressing the hotkey for the bow is an easy way to switch between melee and ranged weapons. Instead of trying to re-equip your one-handed weapon and shield, simply press your bow hotkey once more to switch to the separate left and right hand setup.

As an example I have keys 1, 2 bound to a sword and shield, and key 3 bound to a bow. When getting in melee range I press 3.

The game remembers what where the last left and right hand items/spells equipped when you switch to a Two-Handed weapon. A neat little feature that allows you to only use one hotkey.

If you switch between the Bow and a Two-Handed melee weapon, you only have one hotkey to press, but you can not use this trick. You have to press the distinct hotkey for each weapon. Whereas when you switch between One-Handed + Off-hand and a Two-Handed weapon you can keep pressing the same Two-Handed hotkey to toggle between both setups.

I do not have an answer to the problem but I know what causes it. If you remove your hotkeyed weapon by any means, drop it, store it, rack it etc, and then re-equip it, the hotkey will be screwed forever.

Screwed means the hotkey binding will disappear everytime you use another hotkey. Press Q and you will see the number assigned to it is gone.

You need to get a new weapon if you want the hotkey to work properly again.