Can someone trademark a logo that is well established because it is on a company website and the company did not trademark it? Can someone patent your logo if you did not patent it? Or are all the rights protected because the website has a copyright notice at the bottom of the webpage?
For example Bitcoin logo is not trademarked or patented so does the owner of bitcoin have all rights because bitcoin.org has a copyright notice at the bottom?
Logos cannot be patented. Not by anyone. That ois noty what patents do. Some logos are protected by copyright, many are too simple and do not have the"originality" under US law and the laws of those countries that use a similar standard.
As for trademark, trademark laws vary significantly from one country to another, and the question does not say what country is involved. Also, a mark (which can include a name, a logo, a slogan, and other recognizable words, symbols or images used to identify a product or service) may be protected fully in one country, and not at all in another. This is unlike copyright, where most modern works are protected in all countries.
In some countries there is no protection against competing use without registration. In some, including the US, mere use gives a degree of protection.
It should also be noted that trademarks are usually limited to a category or kind of product. Suppose company Alpha trademarks a purple oval with two red squares as a logo for a brand of beer. Then company Beta wants to use a very similar logo (the oval is slightly more elongated, otherwise the same) for a line of automobiles. No one is going to confuse a car with a beer (I hope), so Beta's use of the mark will not be infringing. There are official lists of categories in each country (many now use an international list), and a trademark registration must specify which categories the mark is for. This is again unlike a copyright. There is an exception for "well-known" or "famous" marks, which get wider protection. One probably cannot use the Coca Cola mark for a line of autos, for example.
In most countries an attempt to register a mark in actual use by a different firm for a competing product will fail if this is noticed by the officials. However, that someone else holds a copyright on the mark (assuming it is copyrightable) will not prevent registration if it is not in use, although any use might be a copyright infringement.
During the registration process there is a chance for the owner of a similar mark to object, and if the objection is accepted the registration fails.The details vary by country.
Under current law, a copyright notice does not grant any rights not already present, and does not grant any rights over content not protectable by copyright. A logo on a page with a copyright notice may or may not be protected by copyright.
For example, consider the well-known logo of Target, a red circle with a smaller white circle in the center. This is an effective logo, but it is too simp0le to be protected by copyright. Any protection it has will come from trademark laws.
Note also that if someone registers a trademark but does not ise it, the registration may be canceled after a time. I think that the time is 5 years in the US. So a "squatter" who registered someone else's mark but did not use it would lose any rights after a limited time. This is again unlike copyright, which lasts for many years once a work is created, whether is is published or used or not.
