Is the description to an anime/manga copyrighted?

For legal questions such as these, it is always good to contact a copyright lawyer in your country. Laws regarding fair use, copyrights and the likes differ per country, and a copyright lawyer should be more than capable of answering your questions. And this answer should not be seen as legal advice.

As Turamarth points out in the comments, the cover image part is covered in my answer Do websites need to pay for using anime posters on their website?.

In general you are free to summarize and create synopsis. If you do so without quoting the original, in most cases you should be ok. As it is not the plot of a story that the copyright applies to, but the composition/arrangement of words.

However if these summaries do make use of original sources material, they could be tested for fair use, and ultimately considered a derivative work. Which can be considered a subsidiary right under the original copyright. Which would require you to obtain a license.

In regards to the 'copy-pasting' from 'a' source with proper mention of it's origin, you will have to look into the licensing of the particular origin you are copy pasting it from. Usually you can find this information at the footer of a page, or on their legal pages. As an example, wikipedia states the following license on the sword art online wiki page

Text is available under the Creative Commons Attribution-ShareAlike License; additional terms may apply. By using this site, you agree to the Terms of Use and Privacy Policy. Wikipedia® is a registered trademark of the Wikimedia Foundation, Inc., a non-profit organization.

Where the Creative Commons Attribution-ShareAlike 3.0 Unported License clearly states you are free to do things with the information under a set of conditions