Atomic orbitals: how is bond strength related to stability, energy, bond length?
There are several factors that affect bond disassociation energies (bond enthalpies) but one quick correlation you can make is bond distance vs bond strength. The interatomic distance of the H2 bond is 74 pm vs 199 pm in Cl2. The H2 bond consists of 1S electrons that are very close to the nucleus, giving a short bond. The valence electrons in Cl2 are 3p (relatively far away). This leads to other properties such as polarizability.

In short your first assumption is incorrect, the trend you are seeing has nothing to do with s or p bonds, but on the bond lengths. Therefore comparing s-s bonds and p-p bonds is meaningless unless you take into account the atomic orbital number and therefore an estimate of bond length. 

For your final questions, the stronger the bond, the more stable, the higher energy, and the shorter the bond. 

For more info this UF link has good information:http://www.science.uwaterloo.ca/~cchieh/cact/c120/bondel.html