Yes you should!
No you should not!
Yes. It's a person's business to do with what they will. In a free society, the free exchange of goods and services is a must. You can't force me to serve or sell to people I don't wish to. Simple. If you support freedom you support this.
Yes, it is your property.
YES. Just as consumers have the right to shop in my store, I have the right to refuse service to said consumer if I don't like x-y-z about them. PRIVATELY held business should have the right to refuse service to anyone for any reason.
Discrimination is a no no. Corporations do not have the right to discriminate anybody; regardless of their race, ethnic, religion, sexuality, disability, ect.
This leads the way to discrimination, which can lead to segregation.