page backgrounds .wrapper

I have a question.

I have a different color for my sidebar. A WP user recommended I change the wrapper for my page instead of the sidebar or widget area alone, in order to keep the theme responsive. There are some pages that do not need that background, so I have added in the css that if the page id=(whatever) then wrapper=none. This is working out ok, but I can’t figure out the page id is for the 404 page or the search results page. I suppose I could put a class into the search page itself if there are no page id numbers for those pages.

—NOTE: I just solved this by adding the code below.
Are there any other pages I’d have to do this with besides search, search results and 404?

.post no-results not-found .wrapper {
    background: none;
}

Also, I am wondering if this is indeed the correct way to do this. Why change the entire page wrapper instead of just having a sidebar background? Pages without sidebars would not include the background obviously. And, if someone else worked on the site and created extra pages, it would not need custom css to eliminate the background when it’s not needed.

Here is the site:
search page – want to eliminate background
https://webseminary.com/?s=CONTACT

faq page – needs background for side menu, as do most pages
https://webseminary.com/faq/

test page – no background needed, therefore must edit stylesheet to remove
https://webseminary.com/test/