Energetic Placement of Atomic Orbitals in the HCl Molecular Orbital Diagram How are the 3p orbitals of chlorine lower in energy than the 1s orbital of hydrogen?
I'd say there are two important things to consider here: Firstly, $\ce{Cl}$ is more electronegative than $\ce{H}$. And secondly, as you proceed from the left to the right in any row of the periodic table, the s and p orbitals go down in energy. This is a consequence of the fact that the valence electrons do not screen each other effectively. Thus, the addition of one proton to the nucleus and one electron does not cancel; instead, the valence electrons "feel" an increased nuclear charge.
The electronegativity difference and the reduced screening among the valence electrons seems to be enough to shift the $3\ce{s}$ and even the $3\ce{p}$ orbital of $\ce{Cl}$ below the energy of the $1\ce{s}$ orbital of $\ce{H}$. The experimental state averaged ionization potentials (IP) could serve as a good guide for the placement of starting orbitals in an orbital interaction diagram. For the $\ce{H}$ $1\ce{s}$ orbital you have an IP of $-13.6 \, \mathrm{eV}$. For the $\ce{Cl}$ $3\ce{s}$ and $3\ce{p}$ orbitals you have IP's of $-25.2 \, \mathrm{eV}$ and $-13.7 \, \mathrm{eV}$, respectively.$^{[1]}$ So, this indicates that the $3\ce{p}$ orbital of $\ce{Cl}$ does indeed lie slightly below the $1\ce{s}$ orbital of $\ce{H}$.

Update:

Here is a table of the experimental state averaged ionization potentials, in electron volts, for the main group atoms. However, those $\ce{p}$ atomic orbitals for groups 1 and 2 and for the $\ce{s}$ and $\ce{p}$ orbitals of the sixth row are not experimentally known and hence, calculated values (indicated by an asterisk) have been used.$^{[2,3]}$ The latter include relativistic corrections. The table is taken from the 2nd Edition of the book Orbital Interactions in Chemistry by T. A. Albright, J. K. Burdett, and M.-H. Whangbo.



[1] L. C. Allen, J. Am. Chem. Soc. 1989, 111, 9003 ; L. C. Allen, J. Am. Chem. Soc. 1992, 114, 1510; L. C. Allen, Acc. Chem. Res. 1990, 23, 175.

[2] A. Vela, J. L. Gazquez, J. Phys. Chem. 1988, 92, 5688.

[3] J. P. Desclaux, At. Nucl. Data Tables 1973, 12, 311.