Is my logo design intellectual property?

You have good leverage to assert your entitlement to payment. As the creator of the website, the contents are copyrightable and since you didn't transfer your interest to the company in writing, it's yours until you decide otherwise.

The logo is more complicated, because if it's the company's trademark, they're the ones who accrue rights to it from use, not you, because it's their trademark, even if they didn't pay for it, and even if you created it. However, you can assert the design copyright rights to it.

If you control the website and can point it elsewhere or shut it down (because you registered it with yourself as registrant or administrative contact, you're in a good position to make demand on this company for full payment.

I can't respond regarding the "other things," since you haven't detailed them.

I suggest you hire an IP lawyer to help you assert and protect your rights. You may have other rights as well, depending on the scope of your oral contract.

Disclaimer: Please note that this answer does not constitute legal advice, and should not be relied on, since each state has different laws, each situation is fact specific, and it is impossible to evaluate a legal problem without a comprehensive consultation and review of all the facts and documents at issue. This answer does not create an attorney-client relationship.

Question: Do I have any rights to the logo?

Answer: Yes. I think you own it outright. A number of very interesting cases have recently held that no trademark rights accrue to the alleged owner of a trademark if the use of the trademark is unlawful. Here, I think payment to you was a condition precedent to the company acquiring any trademark rights in the logo. No payment, no trademark rights via assignment. And using a trademark that you do not own is unlawful -- hence no trademark rights via use.

You have to decide whether this is a battle worth fighting. It will cost you a significant amount to hire an attorney to present your side of the matter. You also do not want to get the reputation in town as hard to deal with. Balance the pros and cons of pursuing the matter and if the pros win out hire an intellectual property attorney. Good luck.