1. An atomic orbital is a region of space around an atom that has a non-zero probability for an electron with a particular energy. Analogously, a molecular orbital is a region of space in a molecule that has a non-zero probability for an electron with a particular energy. Both an atomic orbital and a molecular orbital can contain two electrons.
2. See Ch 9 Figure 2 (Part 1: Molecular Orbital Theory)
3. See Ch 9 Figure 2 (Part 1: Molecular Orbital Theory)
4. No. Because an np_x orbital on one atom is perpendicular to an np_y orbital on an adjacent atom, the net overlap between the two is zero. This is also true for np_y and np_z orbitals on adjacent atoms.
5. The bond order is 1, and the ion has no unpaired electrons
6. The number of molecular orbitals is always equal to the number of atomic orbitals you start with. Thus, combining three atomic orbitals gives three molecular orbitals, and combining four or five atomic orbitals will give four or five molecular orbitals, respectively.
7. 
   
   1. The NO⁺ ion has 10 valence electrons, which fill all the molecular orbitals up to and including the ?_2p. With eight electrons in bonding molecular orbitals and two electrons in antibonding orbitals, the bond order in NO⁺ is (8 ? 2)/2 = 3.
   2. The NO^? ion contains two more electrons, which fill the ??2p molecular orbital. The bond order in NO^? is (8 ? 4)/2 = 2.

                
8.  See answer to 8 here.

9. See the procedures in Part 2: Homonuclear Diatomic Molecules
10.  The p orbitals in this molecule would interact to form 6 MOs (2 ? (bonding and antibonding) and 4 ? (2 bonding and 2 antibonding). We cannot form 3 ? bonding orbitals because the first set of p orbitals will overlap in a head-on fashion creating a sigma bond.