Is there any way you as a pilot can avoid having to deal with the pax/general public?
Sure! Fly cargo!
On any given airliner flight, how much non-luggage cargo is carried, and how much of a flight's revenue generally comes from cargo/mail?
This varies from company to company and the routes that you fly on. Some flights have very little cargo, while others may carry quite a bit of it, particularly mail. I know that UAL used to make quite a bit of money carrying pineapples and mail back to the mainland from Hawaii... more than they made in passenger revenues on the 747. Don't know if they still do.
With regards to logging time, do LSA's count the same as any other GA airplane?
I don't know of any airlines that have started differentiating LSA time. From a hiring standpoint, it's logged as total time, so it can't hurt you, but it may or may not help you.