Answer: 1. In general, Open MPI requires that its executables are in your PATH on every node that you will run on and if Open MPI was compiled as dynamic libraries (which is the default), the directory where its libraries are located must be in your LD_LIBRARY_PATH on every node.
Specifically, if Open MPI was installed with a prefix of /opt/openmpi, then the following should be in your PATH and LD_LIBRARY_PATH
PATH: /opt/openmpi/bin
LD_LIBRARY_PATH: /opt/openmpi/lib
Depending on your environment, you may need to set these values in your shell startup files (e.g., .profile, .cshrc, etc.).
NOTE: there are exceptions to this rule -- notably the --prefix option to mpirun.
See this FAQ entry for more details on how to add Open MPI to your PATH and LD_LIBRARY_PATH.
Additionally, Open MPI requires that jobs can be started on remote nodes without any input from the keyboard. For example, if using rsh or ssh as the remote agent, you must have your environment setup to allow execution on remote nodes without entering a password or passphrase.
Answer: 2. No, but it certainly makes life easier if you do.
A common environment to run Open MPI is in a "Beowulf"-class or similar cluster (e.g., a bunch of 1U servers in a bunch of racks). Simply stated, Open MPI can run on a group of servers or workstations connected by a network. As mentioned above, there are several prerequisites, however (for example, you typically must have an account on all the machines, you can ssh or ssh between the nodes without using a password etc.).
Regardless of whether Open MPI is installed on a shared / networked filesystem or independently on each node, it is usually easiest if Open MPI is available in the same filesystem location on every node. For example, if you install Open MPI to /opt/openmpi-1.3.3 on one node, ensure that it is available in /opt/openmpi-1.3.3 on all nodes.
This FAQ entry has a bunch more information about installation locations for Open MPI.