Can an element have a third energy shell without completely filling the second energy shell?
Yes, it is possible. When an element is excited, some electrons from lower shell jumps to higher shells. For eg, in-ground state suppose an element has electronic configuration
$\ce{(1s)^2 (2s)^2 (2p)^5}$ . If we energize it, then some of the electrons jump up for a short period of time(till the energy is available) . The config becomes $\ce{(1s)^2 (2s)^1(2p)^6 }$.