Is this word only for liquids? Would it be used for running a machine or turning on lights, for example?
It works for machines but lights would be hoʻā.
Most of the definitions in Pukui & Elbert relate to the more normal uses of 'run'
I would have used hoʻokahe but it seems like hoʻoholo works here.
Could this also be translated to "Are you running in the water?"
That would be "Ke holo nei ʻoe i ka wai" . The prefix hoʻo- is a causative prefix, which means you are causing or making something happen. The prompt actually means that you are making the water run (by opening the faucet).