So out of curiosity, are those signs that say "We reserve the right to refuse serve to anyone" legal? If they are, then why would they need to pass this law? If not, does a business have the right under ANY circumstance to refuse service to someone outside of where the law demands it (like a bartender refusing to serve an intoxicated customer)?
It is generally legal to refuse service to anyone, except for certain explicitly protected categories (gender, age, religion etc). The catch is that on the federal level, the said protected category list does not include sexual orientation. In some states, it is protected on state level. In some, it's not protected on state level, but is protected on the municipal level. The latter is especially common in large metro areas (which lean liberal pretty much everywhere) in otherwise conservative states.
So, basically, the reason why that law makes a difference is because it overrides various municipal bylaws prohibiting such discrimination, and carves out a religious exemption that pretty much anyone can arbitrarily claim in practice.
And yes, businesses can arbitrarily discriminate in other ways. They can refuse to serve intoxicated customers, for example, or people with mustaches (though in some cases discriminating based on some trait can be illegal if it is found that in practice it results in clear discrimination against some protected class by virtue of correlation).
Are businesses considered to be public and therefore must be open to everyone or are they considered private and open only to whomever the owner wants (like a private club)?
They can be either, it all depends on how you set things up. If you have some notion of membership, and only serve members, then you can apply pretty much arbitrary filters for people wanting to become members, including those protected classes - i.e. it's perfectly legal to have a whites-only private club, for example (though of course you'll get a lot of flak in the media as soon as it comes out).