How to create low-cost, fire resistant, hidden in-wall storage

Type X drywall, which is typically 5/8" thick (as opposed to 1/2") and contains fiberglass within the gypsum is typically used for fire protection. I believe it increases the burn through time to around 1 hour, and it increases the impact resistance (there's often falling debris in a fire). For further protection, every electrical junction box should be enclosed around the back, top, and bottom with 2x4's and plywood to prevent fires from burning through the box, or a fire that starts in the box from spreading quickly to the other side. Additionally, every penetration in the wall should be sealed with fire rated caulking (it's typically orange in color, and they also have a similar expanding foam of the same color, so it's easy for the building inspector to verify you used the correct product).

The safe and sound insulation is a great addition. The only other thing you could do is to add a 1" air gap and a second wall on your side, which would help with both the spread of fire and sound, but the downside is that you'd lose almost 6" of space in your kitchen.

Realize that no structure will be fire proof, the goal is to slow the spread until the fire department can get there. And the last small fire I saw in a townhouse resulted in half a dozen engines, ladder trucks, etc, because they take the risk of a spreading fire very seriously.

You should have mounting holes in the back of the router. Simply just attach something like a command strip to the wall and hang the router. We use these for the exact purpose in our personal offices at work.