Am I legally allowed to put the
(C) Copyright, all rights reserved
text at the bottom of my website, if I have a logo that I own but have not registered the copyright or trademark in the logo in any way?
You may use the circle c mark (c) normally with a year, the name of the author of the work and possibly the words "all right reserved" without any legal permission.
This gives you more rights legally than you would have from the copyright rights that arise automatically upon the creation of the work.
Registering the copyright with the copyright registrar simply gives you additional litigation rights and must be done before you file a lawsuit to enforce a copyright.
"All rights reserved" means you aren't granting a license to people who see the website to republish it or use it themselves.
"All rights received" would mean that you are using someone else's copyright with their permission, but would more commonly be expressed "used with permission of [name of copyright owner]."
A trademark arises from use in commerce that causes people to associate your mark with your goods and services. You may use the superscript letters TM to claim a common right trademark in a trademarkable item such as a logo or slogan by affixing it to the claimed mark. This is a basis to sue for trademark infringement but requires proof of many elements that can be dispensed with when the Patent and Trademark Office includes a trademark in its principal register following a formal application to them.
You may not say "patent pending" if you have not applied for a patent, may not claim that something is "patented" when a patent has not been approved, and may not use the circle R mark (R) if your claimed trademark has not been including in the principle register of the Patent and Trademark Office.
(This answer is based on U.S. law, but copyright and trademark laws are quite similar on these points internationally.)