In the US companies that deliver goods or services for the public are generally perceived to be protected from litigation since they are considered to be a "common carrier," but this is only the case if they do not discriminate with what things they traffic.

Wrong answer. Facebook has every right in the world to delete content from their site. Research the CDA and Zeran v. AOL.

Would laws like this regarding breastfeeding translate into the online world?

I certainly hope not. The restaurant essentially deprived the child of food, while Facebook simply wouldn't let a woman exhibit her photos on their site.

Also keep in mind that Facebook does allow photos of breastfeeding, so long as the nipple and areola are not visible. It seems to me that Facebook is trying to appeal to the broadest set of users possible, and if they feel that female nipples hinder that, then more power to them. Those people that don't like it can set up their own site and stick the photos there.

Once, a long time ago, people wrote code in assembly... Of course cheap powerful hardware has made that all a thing of the past.

Actually, I would argue that advances in compilers and interpreters have been just as important to that trend as advances in hardware.