1. Businesses are considered private property, even if the space is leased. The landlord, however, can put in clauses in the lease about customer interactions allowed or disallowed, as long as they do not infringe on recognized rights and protected classes. I.e., a business owned by fundie assholes would have to defer to the physical property owner's anti-homophobia clause in their lease contract with the landlord, even in a jurisdiction that does not have anti-homophobic discrimination codes.
2. Businesses in general cannot base their refusal of service on the apparent race, gender, religion or political affiliation of the customer, and in some jurisdictions that also includes sexual orientation. "We reserve the right to refuse service to anyone for any reason" actually has to be based on the customer's bad behavior only, which can include them stating any opinion in a disruptive, loud manner (i.e., drunk asshole or screaming zealot - it disturbs the peace inside the business).
3. In this Colorado case, where the bakery refused to pipe icing words onto a bible-shaped cake spelling out "God Hates Fags" (I paraphrased), well, the owner did offer the customer a prepared piping bag of icing so he could write it himself later. I don't know if Colorado or the town this happened in has LGBT-specific hate crime law (which would not really apply, as this was non-violent), or any codes as to no LGBT discrimination. I have no idea how this incident would play out before a knowledgeable judge. But I do believe that business owners and employees are not slaves to a customer just because money is about to be exchanged for a product and service. If the customer's request is such that the business worries the product will negatively affect their business' reputation, then maybe there is something to consider. Would the bakery have had to ice-on the words, "Hitler was God. Kill all Jews!" - I don't think so.