A lot of questions, judyar. Not all of the houses will be on all of the sites, because not all listings are syndicated to every site. Some of them are not syndicated to any sites.
Zillow, Trulia, Realtor.com, socalmls.com, are all private sites that take listings from various mls's and post them for the public. There might be different homes on each site, but most homes will be on all the sites. There are differences as to how soon the listings show up or are updated, and that is a big deal if you are looking to buy a home.
Example: A home that I had listed and sold is still listed as active on one of these sites, and I am still getting calls about it. It is very common for these sites to have old listings.
I think by public auctions, you mean foreclosure auctions, or Trustee's Sales. Yes, there are good deals to be had at these auctions, but you must pay cash (no financing) and you must check all aspects of the property on your own. You can find these auctions through foreclosure information services like foreclosure radar and realty trac.
The best way to find a home that fits your budget and criteria is to work with a Realtor. We can find all of the available homes in your area and protect your interests when you buy. Using an agent to buy a home doesn't cost you anything, so why are you trying to do it yourself? Please feel free to contact me if you need more information.