hi Mohammad,
If the LRT test is not significant, we ignore M8's sitewise BEB analysis (the sites with omega>1). Saying that another way, we need to see statistical support for the hypothesis that there IS a subset of sites under positive selection (the LRT) before we look at the analysis of which sites those might be (the BEB).
Codeml is indeed comparing the synonymous rate to the non-synonymous rate. Not sure what you meant by that question. You definitely want enough overall divergence (synonymous and non-synonymous) in the alignment to have the power to detect positive selection. Ziheng or someone else may be able to point you towards discussion and publication on how much divergence is useful (too much is not good, too little is not good). We have had good luck with alignments across primates: others use other species sets.
good luck with your analysis,
Janet