What is this supposed to be connected to? It is screwed into the cabin wall. It seems to be a ground cable of some sort.
Also is this hose supposed to be just open like that? This is from the passenger side coming out of the cabin.
Yes to both. The top picture is just a ground to the body from wherever it starts at. You'll find many of them around the car and under the bonnet (hood). Trying to find where it started will be a chore, but shouldn't be too difficult. It's probably on the engine very near by. Use the length of it to search around. You'll know it should connect to something within the arc of its length.
The second picture is the drain for the A/C core. As air goes across it during use, it condenses any water in the air. It drips into a catch basin and drains out through this tube. Without it, water would leak onto your passenger's feet.
The ground wire was attached to the end of the intake manifold. The shiny aluminum part in the center right side of the picture. Repair is needed. The A/C evaporator drain is to be left open.