In a hadoop cluster, should hive be installed on all nodes?

No, it is not something you install on worker nodes. Hive is a Hadoop client. Just run Hive according to the instructions you see at the Hive site.

From Cloudera's Hive installation Guide:

Install Hive on your client machine(s) from which you submit jobs; you do not need to install it on the nodes in your Hadoop cluster.